Right Triangles - SSAT Upper Level Quantitative

By the Theorem:

, and

The perimeter of the parallelogram is

Using the numbers given to us by the question,

units

Use the Pythagorean Theorem to find the length of the hypotenuse.

Use the Pythagorean Theorem to find the hypotenuse.

The distances when put together create a right triangle.

The distance between them will be the hypotenuse or the diagonal side.

You use Pythagorean Theorem or to find the length.

So you plug and for and which gives you,

or .

Then you find the square root of each side and that gives you your answer of .

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.

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 .

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 .

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.

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.

, 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 .

You may recognize these numbers as multiples of 13 and 12 (each by a factor of 3) and remember that sides of length 5, 12 and 13 make a special right triangle. So the other leg would be 15 .

If you don't remember this, you can use Pythagorean theorem:

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

, where and are legs of the triangle and is the hypotenuse.

Plugging in our given values:

Subtracting from each side of the equation:

Taking the square root of each side of the equation:

Simplifying the square root: