Blog Post

Grade 1

Grade 2

Grade 3

Grade 4

Grade 5

Grade 6

Grade 7

Grade 8

Grade 9

Grade 10

Grade 11

Grade 12


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



<ul class="wp-block-list">
<li>Find hypotenuse length in a 45°-45°-90° triangle</li>
</ul>



<h3 class="wp-block-heading">Special Right Triangles&nbsp;</h3>



<p>There are two special <a href="https://www.turito.com/learn/math/right-triangles-grade-9">right triangles</a> with angles measures as 45°, 45°, 90° degrees and 30°, 60°, 90° degrees. The sides of these triangles are in particular ratios and are known as Pythagorean triplets.&nbsp;&nbsp;</p>



<h3 class="wp-block-heading">45°-45°-90° Triangle</h3>



<p>In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is √2</p>



<p>times the length of a leg.&nbsp;</p>



<div class="wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex">
<div class="wp-block-column is-layout-flow wp-block-column-is-layout-flow"><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/2022/09/1-9.png" alt="45°-45°-90° Triangle" class="wp-image-18407" style="width:173px;height:157px" width="173" height="157"/></figure></div></div>



<div class="wp-block-column is-layout-flow wp-block-column-is-layout-flow"><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/2022/09/2-8.png" alt="45°-45°-90° Triangle" class="wp-image-18408" style="width:375px;height:159px" width="375" height="159" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/2-8.png 340w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/2-8-300x127.png 300w" sizes="auto, (max-width: 375px) 100vw, 375px" /></figure></div></div>
</div>



<h3 class="wp-block-heading">Find hypotenuse length in a 45°-45°-90° triangle &nbsp;</h3>



<h4 class="wp-block-heading">Hypotenuse</h4>



<p>A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle.&nbsp;</p>



<p><strong>Example 1:</strong>&nbsp;</p><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>



<p>Find the length of the hypotenuse.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="164" height="158" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2646.png" alt="Example 1: " class="wp-image-18375"/></figure></div>


<p><strong>Solution:</strong>&nbsp;</p>



<p>By the Triangle Sum Theorem, the measure of the third angle must be 45°.&nbsp;&nbsp;</p>



<p>Then the triangle is a 45°-45°-90° triangle, so by theorem, the hypotenuse is</p>



<p>√2 times as long as each leg.&nbsp;</p><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>



<p>Hypotenuse = leg ×√2 45°-45°-90° Triangle Theorem&nbsp;</p>



<p>=&nbsp; 7√2  (Substitute)&nbsp;</p>



<p><strong>Hypotenuse = le</strong>g√𝟐</p>



<p><strong>Example 2:</strong>&nbsp;</p>



<p>Find the length of the hypotenuse.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="193" height="122" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2644.png" alt="Example 2: " class="wp-image-18373"/></figure></div>


<p><strong>Solution:</strong>&nbsp;</p>



<p>By the 45° -45°-90° triangle theorem, the length of the&nbsp; hypotenuse is the length of a leg times √2</p>



<p>hypotenuse = leg × √2</p>



<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;= 3 × √2</p>



<p><strong>The length of the Hypotenuse is 3√2</strong>&nbsp;</p>



<p><strong>Example 3:</strong>&nbsp;</p>



<p>Find leg lengths in a 45°-45°-90° triangle.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="164" height="158" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2646.png" alt="Example 3: " class="wp-image-18376"/></figure></div>


<p><strong>Solution:</strong>&nbsp;</p>



<p>By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a 45°-45°-90° triangle.&nbsp;</p>



<p>hypotenuse = leg × √2       ( 45°-45°-90° Triangle Theorem &nbsp;)</p>



<p><strong>2</strong>√2&nbsp; = <em>x </em>√2            (Substitute) &nbsp;</p>



<p>2√2/√2 = x√2/√2</p>



<p>&nbsp; 2 = x &nbsp;&nbsp; (Simplify)&nbsp;</p>



<p><strong>x = 2</strong>&nbsp;</p>



<p><strong>Example 4:</strong>&nbsp;</p>



<p>Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 10.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="178" height="157" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2650.png" alt="Example 4: " class="wp-image-18385"/></figure></div>


<p><strong>Solution:</strong>&nbsp;</p>



<p>hypotenuse = leg ×√2</p>



<p>10 = √x&nbsp;&nbsp;×2</p>



<p>x =&nbsp;10/√2</p>



<p>x = 10/√2 * √2/√2</p>



<p>x = 10*√2/2&nbsp;</p>



<p><strong>x = 5 √2</strong></p>



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



<ul class="wp-block-list" type="1">
<li>Identify the legs and hypotenuse of the triangle.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="236" height="232" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2652.png" alt="Identify the legs and hypotenuse of the triangle." class="wp-image-18392"/></figure></div>


<ul class="wp-block-list">
<li>45°-45°-90° TRIANGLES: Find the value of x. Write your answer in the simplest radical form.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="283" height="194" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2653.png" alt="45°-45°-90° TRIANGLES: Find the value of x. Write your answer in the simplest radical form." class="wp-image-18393"/></figure></div>


<ul class="wp-block-list">
<li>The square tile shown has painted corners in the shape of congruent 45°-45°-90° triangles. What is the value of x? What is the side length of the tile?</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="278" height="194" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2655.png" alt="The square tile shown has painted corners in the shape of congruent 45°-45°-90° triangles. What is the value of x? What is the side length of the tile?" class="wp-image-18395"/></figure></div>


<ul class="wp-block-list">
<li>Copy and complete the table.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="224" height="213" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2654.png" alt="Copy and complete the table." class="wp-image-18394"/></figure></div>

<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="466" height="155" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2656.png" alt="Copy and complete the table." class="wp-image-18397" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2656.png 466w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2656-300x100.png 300w" sizes="auto, (max-width: 466px) 100vw, 466px" /></figure></div>


<ul class="wp-block-list">
<li>Is it possible to build a triangle using the given side lengths?</li>
</ul>



<p>4, 4, and 7</p>



<ul class="wp-block-list">
<li>Find the values of the variable(s). Write your answer(s) in the simplest radical form.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="267" height="180" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2657.png" alt="Find the values of the variable(s). Write your answer(s) in the simplest radical form." class="wp-image-18398"/></figure></div>


<ul class="wp-block-list">
<li>Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 6.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="227" height="195" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2658.png" alt="Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 6." class="wp-image-18399"/></figure></div>


<ul class="wp-block-list">
<li>Find the length of the hypotenuse in the 45° – 45° – 90° triangle. Write your answer in radical form.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="220" height="195" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2659.png" alt="Find the length of the hypotenuse in the 45° – 45° – 90° triangle. Write your answer in radical form." class="wp-image-18400"/></figure></div>


<ul class="wp-block-list">
<li>Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="244" height="210" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2660.png" alt="Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22." class="wp-image-18401"/></figure></div>


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


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="278" height="165" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2661.png" alt="Find the length of the hypotenuse." class="wp-image-18403"/></figure></div>


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


<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-2651.png" alt="Concept Map" class="wp-image-18390" style="width:539px;height:323px" width="539" height="323" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2651.png 885w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2651-300x180.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2651-768x461.png 768w" sizes="auto, (max-width: 539px) 100vw, 539px" /></figure></div>


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



<ul class="wp-block-list">
<li>Understand <a href="https://www.turito.com/learn/math/special-right-triangles-grade-9" data-type="link" data-id="https://www.turito.com/learn/math/special-right-triangles-grade-9">special right triangles</a>.</li>



<li>Understand 45°- 45° -90°&nbsp; Triangle theorem</li>



<li>Understand how to find the length of the hypotenuse.</li>



<li>Understand&nbsp; how to find the lengths of the legs in the triangle.</li>
</ul>
