Prove relationships in geometric figures - HiSet: High School Equivalency Test: Math

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ and $\angle B \cong \angle E$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Answer

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 $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\angle C \cong \angle F$ and $\angle A \cong \angle D$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Answer

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 $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ and $\angle C \cong \angle F$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Answer

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 $\bigtriangleup ABC \sim \bigtriangleup DEF$ ; 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.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Answer

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.

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Question

$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 4:5, with $\bigtriangleup ABC$ the smaller triangle.

Complete the sentence: the perimeter of $\bigtriangleup ABC$ is _______% less than that of $\bigtriangleup DEF$ .

(Select the closest whole percent)

Answer

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is $\frac{4}{5}$ .

This makes the perimeter of the smaller triangle $\bigtriangleup ABC$ equal to $\frac{4}{5} \times 100 % = 80 %$ of that of larger triangle $\bigtriangleup DEF$ —or, equivalently, less.

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Question

$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 5:4, with $\bigtriangleup ABC$ the larger triangle.

Complete the sentence: the perimeter of $\bigtriangleup ABC$ is _______% greater than that of $\bigtriangleup DEF$ .

(Select the closest whole percent).

Answer

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is $\frac{5}{4}$ .

This makes the perimeter of larger triangle $\bigtriangleup ABC$ equal to $\frac{5}{4} \times 100 % = 125 %$ of that of smaller triangle $\bigtriangleup DEF$ —or, equivalently, 25% greater.

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Question

$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 4:5, with $\bigtriangleup ABC$ the smaller triangle.

Complete the sentence: the area of $\bigtriangleup ABC$ is _______% less than that of $\bigtriangleup DEF$ .

(Select the closest whole percent).

Answer

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to $\frac{4} {5}$ , so the ratio of the areas is the square of this, or $\left ( \frac{4}{5} \right ) ^{2} = \frac{16}{25}$ .

This makes the area of smaller triangle $\bigtriangleup ABC$ equal to $\frac{16}{25} \times 100 % = 64 %$ of that of larger triangle $\bigtriangleup DEF$ —or, equivalently, less.

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Question

$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 5:4, with $\bigtriangleup ABC$ the larger triangle.

Complete the sentence: the area of $\bigtriangleup ABC$ is _______% greater than that of $\bigtriangleup DEF$ .

(Select the closest whole percent)

Answer

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to $\frac{5}{4}$ , so the ratio of the areas is the square of this, or $\left ( \frac{5}{4} \right ) ^{2} = \frac{25}{16}$ .

This makes the area of larger triangle $\bigtriangleup ABC$ equal to $\frac{25}{16} \times 100 % = 156 \frac{1}{4} %$ of that of smaller triangle $\bigtriangleup DEF$ —or, equivalently, $56 \frac{1}{4} %$ greater.

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Question

A triangle has sides of length 8 and 12; the triangle is scalene and obtuse. Which of the following could be the length of its third side?

Answer

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,

$a^{2}+b^{2} < c^{2}$ .

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

$10^{2}+ 8^{2}= 100 + 64 = 164 > 144 = 12^{2}$ ,

violating this condition.

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

$6^{2}+ 8^{2}= 36 + 64 = 100 < 144 = 12^{2}$ ,

satisfying this condition.

This makes 6 the correct choice.

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