Question

analyzing a method
a student broke this trapezoid into two triangles. does the student have enough information to determine the areas of both triangles?
yes, one triangle has a base of 6 in. and the other has a base of 4 in. both have a height of 5 in.
no, each triangle has a height of 5 in. but the bases of the triangles cannot be determined.
no, one triangle has a base of 6 in. and the other has a base of 4 in. but the heights of the triangles cannot be determined.
yes, each triangle has a base of 6 in. one triangle has a height of 4 in. and the other has a height of 5 in.

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the height.

Step2: Analyze the trapezoid - to - triangle transformation

When a trapezoid is divided into two triangles by a diagonal, the height of each triangle is the same as the height of the trapezoid. In this case, the height of the trapezoid is \(h = 5\) inches.
If we assume the bases of the two triangles formed from the trapezoid (with parallel sides \(a = 4\) inches and \(b=6\) inches) are \(b_1 = 4\) inches and \(b_2=6\) inches respectively.
For the first triangle with \(b_1 = 4\) inches and \(h = 5\) inches, the area \(A_1=\frac{1}{2}\times4\times5=10\) square inches.
For the second triangle with \(b_2 = 6\) inches and \(h = 5\) inches, the area \(A_2=\frac{1}{2}\times6\times5 = 15\) square inches.

Answer:

Yes, one triangle has a base of 6 in. and the other has a base of 4 in. Both have a height of 5 in.