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<h2 class="wp-block-heading">Key Concepts</h2>



<ul class="wp-block-list"><li>Find the length of a hypotenuse</li><li>Verify right triangles</li></ul>



<h2 class="wp-block-heading">The Pythagorean Theorem </h2>



In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2601.png" alt="The Pythagorean Theorem " class="wp-image-18298" width="646" height="228" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2601.png 889w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2601-300x106.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2601-768x272.png 768w" sizes="auto, (max-width: 646px) 100vw, 646px" /></figure></div>


<h3 class="wp-block-heading">Find the length of a hypotenuse </h3>



Hypotenuse: A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle.&nbsp;



The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.&nbsp;



Example 1:&nbsp;



Find the length of the hypotenuse of the right triangle.&nbsp;<div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes" data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="165" height="129" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2597.png" alt="Example 1:" class="wp-image-18294"/></figure></div>


Solution:&nbsp;



(hypotenuse)2   = (leg)2 + (leg)2      (Pythagorean Theorem) 



x2 = 62 +42 (Substitute) 



x2 =36 +16 (Multiply) 



x2 = 52 (Add) <div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes" data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



x = (52 )1/2 (Radical form) 



Example 2:&nbsp;



Ria ties an apple balloon to a stake in the ground. The rope is 10 feet long. As the wind picks up, Ria observes that the balloon is now 6 feet away from the stake. How far above the ground is the balloon now?&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2603.png" alt="Example 2: " class="wp-image-18300" width="337" height="318" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2603.png 466w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2603-300x283.png 300w" sizes="auto, (max-width: 337px) 100vw, 337px" /></figure></div>


Solution:&nbsp;



(Length of rope)2&nbsp; =&nbsp; (Distance from stake)2&nbsp; + (Height of rope)2&nbsp;



102  = 62  + x 2 (Substitute)



100 = 36 + x 2 (Multiply) 



64 = x 2 (Subtract 64 from each side) 



√64  = x (Find positive square root) 



8 = x&nbsp;



The balloon is 8 ft above the ground.&nbsp;



Example 3:&nbsp;



Find the area of the isosceles triangle with side lengths 30 ft., 18 ft., and 18 ft. &nbsp;



Solution:&nbsp;



STEP 1:&nbsp;



Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown.&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2598.png" alt="STEP 1: " class="wp-image-18295" width="310" height="158"/></figure></div>


STEP 2:&nbsp;



Use the Pythagorean Theorem to find the height of the triangle.&nbsp;



C2 =   a2 + b2 (Pythagorean Theorem) 



182 = 152 + h2 (Substitute) 



324 = 225 + h2 (Multiply)



99 = h2 (Subtract 225 from each side)  



9.9=h&nbsp;



STEP 3: 



Find the area.&nbsp;



&nbsp;Area = 1/2 (base) (height)&nbsp;



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = 1/2 (30) (9.9)&nbsp;



            = 148.1 ft. 2 



EXAMPLE 3:&nbsp;



Find the length of the hypotenuse of the right triangle.&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="161" height="135" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2599.png" alt="" class="wp-image-18296"/></figure></div>


Solution:&nbsp;



Method 1:&nbsp;&nbsp;



Use a Pythagorean triple.&nbsp;



A common Pythagorean triple is 3, 4 , 5.&nbsp;&nbsp;



Notice that if you multiply the lengths of the legs of the Pythagorean triple by 3, you get the lengths of the legs of this triangle:&nbsp;&nbsp;



3 × 3 = 9 and 



4 × 3 =12&nbsp;



So, the length of the hypotenuse is&nbsp;&nbsp;



5 × 3 = 15&nbsp;



Method 2: 



Method 2: Use the Pythagorean Theorem.&nbsp;



 x2   = 92  +122 



x2   = 81 + 144 



x2   = 225 



x = 15 



<h3 class="wp-block-heading">Use the Converse of the Pythagorean Theorem </h3>



The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="414" height="125" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2600.png" alt="Use the Converse of the Pythagorean Theorem " class="wp-image-18297" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2600.png 414w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2600-300x91.png 300w" sizes="auto, (max-width: 414px) 100vw, 414px" /></figure></div>


&nbsp;Then ΔABC is a right triangle.&nbsp;



<h3 class="wp-block-heading">Verify right triangles </h3>



Example 1: 



Write whether the given triangle is a right triangle.&nbsp;


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="143" height="116" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2602.png" alt="Example 1: " class="wp-image-18299"/></figure></div>


Solution:&nbsp;



Let c represent the length of the longest side of the triangle.&nbsp;&nbsp;



Check to see whether the side lengths satisfy the equation c2 = a2 + b2 ? 



972   = 652 + 722? 



9,409 = 4,225 + 5,184&nbsp;



9,409 = 9,409 



The triangle is a right triangle.&nbsp;



<h2 class="wp-block-heading">Exercise</h2>



<ul class="wp-block-list" type="1"><li>Find the length of the hypotenuse of the right triangle.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="261" height="135" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2605.png" alt="Find the length of the hypotenuse of the right triangle." class="wp-image-18302"/></figure></div>

<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="228" height="150" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2606.png" alt="Find the length of the hypotenuse of the right triangle." class="wp-image-18304"/></figure></div>


<ul class="wp-block-list" type="1"><li>Find the unknown length of x.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2607.png" alt="Find the unknown length of x." class="wp-image-18305" width="176" height="202" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2607.png 265w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2607-262x300.png 262w" sizes="auto, (max-width: 176px) 100vw, 176px" /></figure></div>


<ul class="wp-block-list"><li>What is the area of a right triangle with a leg length of 15 feet and a hypotenuse length of 39 feet?</li></ul>



A) 270 ft2 &nbsp;



B) 292.5 ft2



C) 540 ft2 &nbsp;&nbsp;



D) 585 ft2



<ul class="wp-block-list"><li>Tell whether a triangle with the given side lengths is a right triangle.</li></ul>



3.5, √61



<ul class="wp-block-list" type="1"><li>Tell whether the triangle is a right triangle.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="267" height="150" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2608.png" alt="Tell whether the triangle is a right triangle." class="wp-image-18307"/></figure></div>

<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="272" height="165" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2609.png" alt="Tell whether the triangle is a right triangle." class="wp-image-18308"/></figure></div>


<ul class="wp-block-list"><li>A student tells you that if you double all the sides of a right triangle, the new triangle is&nbsp; obtuse. Explain why this statement is incorrect.</li><li>You are making a canvas frame for a painting using stretcher bars. The rectangular shape painting will be 10 inches long and 8 inches wide. With a ruler, how can you be certain that the corners of the frame are 90°?</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="268" height="170" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2610.png" alt="You are making a canvas frame for a painting using stretcher bars. The rectangular shape painting will be 10 inches long and 8 inches wide. With a ruler, how can you be certain that the corners of the frame are 90°?" class="wp-image-18311"/></figure></div>


<ul class="wp-block-list"><li>Tell whether the given side lengths of a triangle can represent a right triangle.</li><li>9, 12, and 15</li><li>36, 48, and 60</li><li>Decide if the segment lengths form a triangle. Find whether the triangle is acute, right, or obtuse?</li><li>12, 16, and 20</li><li>15, 20, and 36</li><li>What type of triangle has side lengths of 4, 7, and 9?</li></ul>



a.  Acute scalene



b.  Right scalene



c.  Obtuse scalene 



d.  None of the above



<h3 class="wp-block-heading">Concept Map</h3>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="1000" height="405" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2604.png" alt="Concept Map" class="wp-image-18301" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2604.png 1000w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2604-300x122.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2604-768x311.png 768w" sizes="auto, (max-width: 1000px) 100vw, 1000px" /></figure></div>


<h3 class="wp-block-heading">What have we learned</h3>



<ul class="wp-block-list"><li>Understand Pythagorean theorem</li><li>Understand how to find the length of a hypotenuse</li><li>Understand converse of the Pythagorean theorem</li><li>Verify right triangles.</li></ul>