Question

Shonda is asked to draw a triangle with the following specifications: two angles are complementary two angles have equal measure the shortest side is half the length of the longest side Which of the following statements about this triangle is true? A. More than one triangle exists with the given conditions, and all instances must be isosceles triangles. B. Exactly one triangle exists with the given conditions, and it must be a right isosceles triangle. C. More than one triangle exists with the given conditions, and all instances must be right triangles. D. No triangle exists with the given conditions.

Answer 1

"two angles are complementary;" that means the sum of two angles is 90.  Since the sum of all three is 180, the last angle is 90.  It is a right triangle.

"two angles have equal measure;" we can't have two angles which are 90, so the other two are equal.  But they sum to 90, so what number plus itself is 90?  45+45=90.  We know now that the triangle has measures 45, 45, and 90.

We know that the legs of the triangle are the same, so we can use the Pythagorean theorem to show that $h^2=2l^2$, where h is the length of the hypotenuse and l is the length of the legs.

Of course, this means $h=l\sqrt2$.  But "the shortest side is half the length of the longest side"!  No triangles like this exist.

The only true statement is D