Prove relationships in geometric figures - HiSET

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that .

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. If the angles were the included angles of the triangles, then the SAS Similarity Theorem could be applied to prove that ; however, the two congruent angles are nonincluded, and there is no "SSA" statement that can be applied to prove similarity. Without further information, it cannot be proved that the triangles are similar.

We are given that, between the triangles, two pairs of corresponding sides are proportional. Without knowing anything else, the proportionality of two pairs of sides is insufficient to prove that the triangles are similar.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of the smaller triangle equal to of that of larger triangle —or, equivalently, less.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of larger triangle equal to of that of smaller triangle —or, equivalently, 25% greater.

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of smaller triangle equal to of that of larger triangle —or, equivalently, less.

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of larger triangle equal to of that of smaller triangle —or, equivalently, greater.

A scalene triangle has three sides of different lengths. The triangle is known to have sides of length 8 and 12, so this eliminates 8 and 12 as the correct choices for the length of the third side.

The sum of the lengths of the two smallest sides must exceed the length of the third side. 4 can be eliminated as a the correct choice, since , violating this condition.

This leaves 6 and 10 as possible answers. For a triangle to be obtuse, it must hold that if are its sidelengths, the greatest of the three,

.

If the length of the third side is 10, setting , we see that

,

violating this condition.

If the length of the third side is 6, setting , we see that

,

satisfying this condition.

This makes 6 the correct choice.

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. The angles that are congruent are the included angles of their respective sides. By the SAS Similarity Postulate, this is enough to prove that .

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that .

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. If the angles were the included angles of the triangles, then the SAS Similarity Theorem could be applied to prove that ; however, the two congruent angles are nonincluded, and there is no "SSA" statement that can be applied to prove similarity. Without further information, it cannot be proved that the triangles are similar.

We are given that, between the triangles, two pairs of corresponding sides are proportional. Without knowing anything else, the proportionality of two pairs of sides is insufficient to prove that the triangles are similar.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of the smaller triangle equal to of that of larger triangle —or, equivalently, less.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of larger triangle equal to of that of smaller triangle —or, equivalently, 25% greater.

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of smaller triangle equal to of that of larger triangle —or, equivalently, less.

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of larger triangle equal to of that of smaller triangle —or, equivalently, greater.

A scalene triangle has three sides of different lengths. The triangle is known to have sides of length 8 and 12, so this eliminates 8 and 12 as the correct choices for the length of the third side.

The sum of the lengths of the two smallest sides must exceed the length of the third side. 4 can be eliminated as a the correct choice, since , violating this condition.

This leaves 6 and 10 as possible answers. For a triangle to be obtuse, it must hold that if are its sidelengths, the greatest of the three,

.

If the length of the third side is 10, setting , we see that

,

violating this condition.

If the length of the third side is 6, setting , we see that

,

satisfying this condition.

This makes 6 the correct choice.

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. The angles that are congruent are the included angles of their respective sides. By the SAS Similarity Postulate, this is enough to prove that .

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that .