Properties of Triangles - SSAT Upper Level: Quantitative

Since an equilateral triangle has sides that are all the same, divide the perimeter by to get the length of each side.

Since an equilateral triangle has three equal sides, divide the perimeter by to find the length of each side.

Since an equilateral triangle has three equal sides, divide the perimeter by to find the length of one side.

Of the choices listed, the main difference between an equilateral triangle and a scalene triangle is their side lengths. An equilateral triangle has to have equal sides, but a scalene triangle can have all different side lengths.

Since an equilateral triangle has side lengths that are all the same, multiply the length of one side by to find the perimeter.

The total number of degrees in a triangle is .

While is provided as the measure of one of the angles in the diagram, you are also told that the triangle is a right triangle, meaning that it must contain a angle as well. To find the value of , subtract the other two degree measures from .

One of the angles of a right triangle is by definition a right, or , angle, so this is the measure of one of the missing angles. Since the measures of the angles of a triangle total , if we let the measure of the third angle be , then:

The other two angles measure .

A right triangle must have one right angle and two acute angles; this means that no angle of a right triangle can be obtuse. But since , it is obtuse. This makes it impossible for a right triangle to have a angle.

By definition, a right triangle has to have one right angle, or a angle, and an isosceles triangle has equal base angles and two equal side lengths.

All the angles in a triangle must add up to 180 degrees.

All the angles in a triangle adds up to .

All the angles in a triangle add up to degrees.

All the angles in a triangle add up to .

, and corresponding parts of congruent triangles are congruent.

Since is a right angle, so is . and ; since , it follows that . is an isosceles right triangle; consequently, .

is a 45-45-90 triangle with hypotenuse of length . By the 45-45-90 Triangle Theorem, the length of each leg is equal to that of the hypotenuse divided by ; therefore,

is eliminated as the correct choice.

Also, the perimeter of is

.

This eliminates the perimeter of being 40 as the correct choice.

Also, is eliminated as the correct choice, since the triangle is 45-45-90.

The area of is half the product of the lengths of its legs:

The correct choice is the statement that has area 100.

; since both are right angles.

Given that two pairs of corresponding angles are congruent and any one side of corresponding sides is congruent, it follows that the triangles are congruent. In the case of Statement I, the included sides are congruent, so by the Angle-Side-Angle Congruence Postulate, . In the case of the other two statements, a pair of nonincluded sides are congruent, so by the Angle-Angle-Side Congruence Theorem, . Therefore, the correct choice is I, II, or III.

has as its leg lengths 10 and 24, so the length of its hypotenuse, , is

Its perimeter is the sum of its sidelengths:

Its area is half the product of the lengths of its legs:

and have the same perimeter and area, respectively, as ; also, between and , corresponding angles are congruent. In the absence of other information, none of these three triangles can be eliminated as being congruent to .

However, and . Therefore, . Since a pair of corresponding sides is noncongruent, it follows that .

For the triangles to be similar, the dimensions of all sides must have the same ratio by dividing the 3-4-5 triangle.

The 6-8-10 triangle will have a scale factor of 2 since all dimensions are doubled the original 3-4-5 triangle.

The only correct answer that will yield similar ratios is the triangle with a scale factor of 4 from the 3-4-5 triangle.

The other answers will yield different ratios.

We are looking for a Pythagorean triple - that is, three integers that satisfy the relationship . We know that , and the only Pythagorean triple with is . The legs of the triangle are therefore 5 and 12, and the area of the right triangle is

The area of a right triangle is half the product of its legs. The area of Right Triangle A is equal to square inches; that of Right Triangle B is equal to square inches. The sum of the areas is square inches, which is the area of Rectangle C.

The area of a rectangle is the product of its length and its height. Therefore, the height is the quotient of the area and the length, which, for Rectangle C, is inches.