Question

Amira is writing a coordinate proof to show that the area of a triangle created by joining the midpoints of an isosceles triangles is one-fourth the area of the isosceles triangle. She starts by assigning coordinates as given.
Triangle D E F in the coordinate plane so that vertex D is at the origin and is labeled 0 comma 0, vertex E is in the first quadrant and is labeled 2 a comma 2 b, and vertex F is on the positive side of the x-axis and is labeled 4a comma 0. Point Q is between points D and E. Point R is between points E and F. Point P is between points D and F and is labeled 2 a comma 0.

Enter your answers, in simplest form, in the boxes to complete the coordinate proof.
Point Q is the midpoint of DE¯¯¯¯¯ , so the coordinates of point Q are (a, b) .

Point R is the midpoint of FE¯¯¯¯¯ , so the coordinates of point R are (
, b).

In △DEF , the length of the base, DF¯¯¯¯¯ , is
, and the height is 2b, so its area is
.

In △QRP , the length of the base, QR¯¯¯¯¯ , is
, and the height is b, so its area is ab .

Comparing the expressions for the areas proves that the area of the triangle created by joining the midpoints of an isosceles triangle is one-fourth the area of the

Answer 1

Answer: We can fill the boxes with help of below explanation.

Explanation: According to the given figure,

Triangle DEF is an isosceles triangle with sides DE, EF and FD.

And, where  DE=EF and P,R and Q are the midpoints of the sides DF, EF and DE respectively.

Theorem: Area of a triangle created by joining the midpoints of an isosceles triangles is one-fourth the area of the isosceles triangle.

Proof:

Point Q is the midpoint of DE, so the coordinates of point Q are (a, b). (because mid point of a line segment always has coordinates half of the sum of corresponding coordinates of end points. )

Point R is the midpoint of FE , so the coordinates of point R are (

3a,b).

In triangle DEF , the length of the base, DF is  4a and the height is 2b, So, its area is 4ab.( because, Area of a triangle= 1/2×base×height

In triangle QRP , the length of the base, QR is   2a and the height is b,

So, its area is ab.

On comparing the above two expression, we found that, area of triangle DEF =4( area of triangle QRP)

Answer 2

Point R is the midpoint of FE¯¯¯¯¯ , so the coordinates of point R are (3a, b).

In △DEF , the length of the base, DF¯¯¯¯¯ , is
4a, and the height is 2b, so its area is
1/2×4a×2b = 4ab.

In △QRP , the length of the base, QR¯¯¯¯¯ , is
3a-a = 2a, and the height is b, so its area is 1/2×2a×b = ab .

Comparing the expressions for the areas proves that the area of the triangle created by joining the midpoints of an isosceles triangle is one-fourth the area of the larger isosceles triangle.