Maths-
General
Easy
Question
State and prove the Midsegment Theorem.
The correct answer is: Hence proved
SOLUTION:
HINT: State and prove the theorem.
Complete step by step solution:
Triangle midsegment theorem states that the line segment connecting the midpoints
of any 2 sides of a triangle is
- Is one half the length of the third side.
- Is parallel to the third side.
Proof:
In △ABC, we connect 2 midpoints D and E of 2 sides AB and BC.
Consider 2 triangles, △ABC and △DBE
We have,
AD = BD (since D is the midpoint)
CE = BE (since E is the midpoint)
and ∠ABC = ∠DBE by SAS similarity criterion.
⇒ △ ABC and △DBE are similar triangles.
We know that corresponding sides of similar triangles are proportional.
Here, we have proved the first part.
Now, Take 2 line segments DE and AC and a transversal BC cutting these 2 lines.
Since, △ABC and △DBE are similar triangles we have congruent corresponding
angles.
⇒ ∠BED corresponds to ∠BCA
⇒ ∠BED ⩭ ∠BCA
So by the converse of corresponding angles theorem, since a pair of corresponding
angles created by the transversal BC are congruent, we conclude that DE and AC
are parallel.
Hence proved.
Related Questions to study
Maths-
Find the midpoint and length of each side using the placement
below. What is the advantage of using 2h instead of h for the leg lengths?
Find the midpoint and length of each side using the placement
below. What is the advantage of using 2h instead of h for the leg lengths?
Maths-General
Maths-
A sum of Rs. 4,000 is lent for 5 years at the rate of 15% per annum. Find the simple interest.?
A sum of Rs. 4,000 is lent for 5 years at the rate of 15% per annum. Find the simple interest.?
Maths-General
Maths-
Place a rectangle in a coordinate plane in a way that is
convenient for finding side lengths. Assign coordinates to each vertex.
Place a rectangle in a coordinate plane in a way that is
convenient for finding side lengths. Assign coordinates to each vertex.
Maths-General
Maths-
MN is the midsegment of △ ABC.
Find MN.
Also, find AC if CN = 35 cm
MN is the midsegment of △ ABC.
Find MN.
Also, find AC if CN = 35 cm
Maths-General
Maths-
In △ ABC, AM ≅ BM
BQ ≅ QC
Prove that MQ ∥ AC.
In △ ABC, AM ≅ BM
BQ ≅ QC
Prove that MQ ∥ AC.
Maths-General
Maths-
Write a coordinate proof.
Any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles.(Hint: Draw the segment from the right angle to the midpoint of the hypotenuse.)
Write a coordinate proof.
Any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles.(Hint: Draw the segment from the right angle to the midpoint of the hypotenuse.)
Maths-General
Maths-
NQ, MN and MQ are the midsegments of △ ABC. Find BM.
NQ, MN and MQ are the midsegments of △ ABC. Find BM.
Maths-General
Maths-
MN is the midsegment of △ ABC.
Find MN if BC = 35 m.
MN is the midsegment of △ ABC.
Find MN if BC = 35 m.
Maths-General
Maths-
MN is the midsegment of △ ABC.
Find BC if MN = 17 cm.
MN is the midsegment of △ ABC.
Find BC if MN = 17 cm.
Maths-General
Maths-
In the figure above, RT = TU. What is the value of x?
In the figure above, RT = TU. What is the value of x?
Maths-General
Maths-
For the graph below , write an inequality and explain the reasoning :
Note:
We do not use < or > for infinity in inequalities, since the variable extends to infinite and has no real fixed value.
For the graph below , write an inequality and explain the reasoning :
Maths-General
Note:
We do not use < or > for infinity in inequalities, since the variable extends to infinite and has no real fixed value.
Maths-
In what time will Rs 10000 earn an interest of Rs. 12600 at 18% per annum simple interest?
In what time will Rs 10000 earn an interest of Rs. 12600 at 18% per annum simple interest?
Maths-General
Maths-
The formula below is often used by project managers to compute E, the estimated time to complete a job, where O is the shortest completion time, P is the longest completion time, and M is the most likely completion time.
Which of the following correctly gives P in terms of E, O, and M?
The formula below is often used by project managers to compute E, the estimated time to complete a job, where O is the shortest completion time, P is the longest completion time, and M is the most likely completion time.
Which of the following correctly gives P in terms of E, O, and M?
Maths-General
Maths-
A square has vertices (0, 0), (3, 0), and (0, 3). Find the fourth
Vertex.
A square has vertices (0, 0), (3, 0), and (0, 3). Find the fourth
Vertex.
Maths-General
Maths-
Let a, b and c be real numbers, c≠0, Show that each of the following statements is true :
1. If a>b and c<0 , then ca< cb
2. If a>b and c<0 , then .
Note:
Whenever we multiply or divide an inequality by a negative number, the direction of the inequality symbol changes.
Let a, b and c be real numbers, c≠0, Show that each of the following statements is true :
1. If a>b and c<0 , then ca< cb
2. If a>b and c<0 , then .
Maths-General
Note:
Whenever we multiply or divide an inequality by a negative number, the direction of the inequality symbol changes.
