1.
Which of the following triangles are always similar?
Correct Answer
C. Equilateral triangle
Explanation
Equilateral triangles are always similar because all three sides are equal in length and all three angles are equal. This means that if you have two equilateral triangles, you can scale one up or down and rotate it, and it will still be congruent to the other triangle. Therefore, all equilateral triangles are similar to each other.
2.
The sides of a triangle are 7, 10, and 12. Find the length of the longest side of a similar triangle whose shortest side is 21.
Correct Answer
D. 36
Explanation
In a similar triangle, the corresponding sides are proportional. To find the length of the longest side of the similar triangle, we can set up a proportion using the given lengths of the sides.
Let x be the length of the longest side of the similar triangle.
Using the proportion:
7/21 = 12/x
Cross-multiplying:
7x = 21 * 12
Simplifying:
7x = 252
Dividing both sides by 7:
x = 36
Therefore, the length of the longest side of the similar triangle is 36.
3.
In triangle ABC, angle A = 90º and angle B = 25º. In triangle DEF, angle E = 25º and angle F = 65º. Are the triangles similar?
Correct Answer
B. True
Explanation
Yes, the triangles are similar. Since angle B in triangle ABC is equal to angle E in triangle DEF, and angle A in triangle ABC is equal to angle F in triangle DEF, by the Angle-Angle (AA) similarity criterion, the triangles are similar.
4.
Two triangles are similar. If the ratio of the perimeters is 5:3, find the ratio of the corresponding sides.
Correct Answer
A. 5:3
Explanation
If two triangles are similar, it means that their corresponding angles are equal, and the ratio of their corresponding sides is the same. In this case, the ratio of the perimeters of the triangles is given as 5:3. The perimeter of a triangle is the sum of its three sides. Since the ratio of the perimeters is 5:3, it means that the sum of the corresponding sides of the triangles is also in the ratio of 5:3. Therefore, the ratio of the corresponding sides of the triangles is 5:3.
5.
Find the unknown length x in the following figure.
Correct Answer
15
6.
Which of the following conditions are NOT sufficient to identify a pair of congruent triangles?
Correct Answer
C. Two pairs of corresponding sides are proportional and one pair of non-included angles are equal (SSA).
Explanation
The condition that is NOT sufficient to identify a pair of congruent triangles is "Two pairs of corresponding sides are proportional and one pair of non-included angles are equal (SSA)." While the Side-Side-Angle (SSA) condition may suggest similarity, it alone is not enough to guarantee congruence between triangles.
7.
What is true of angles of similar triangles?
Correct Answer
B. Corresponding angles are congurent.
Explanation
In similar triangles, corresponding angles are congruent. This means that if two triangles are similar, the corresponding angles in each triangle have the same measure. This property is a key characteristic of similar triangles and is used to determine if two triangles are similar or not. Opposite angles being congruent, corresponding angles being 90 degrees, and corresponding angles being parallel are not true for all similar triangles.
8.
Area of an equilateral triangle with side length a is equal to:
Correct Answer
C. √3/4 a^{2}
Explanation
The correct answer, √3/4 a^2, represents the formula for finding the area of an equilateral triangle. In this formula, "a" represents the length of one side of the triangle. The square of "a" (a^2) is multiplied by the constant √3/4 to find the area. This formula is derived from the geometric properties of equilateral triangles and is a commonly used formula in mathematics.
9.
If the perimeter of a triangle is 100 cm and the length of the two sides are 30 cm and 40 cm, the length of the third side will be:
Correct Answer
A. 30
Explanation
The perimeter of a triangle is the sum of the lengths of all its sides. In this case, the perimeter is given as 100 cm. The length of the two sides are given as 30 cm and 40 cm. To find the length of the third side, we subtract the sum of the lengths of the two given sides from the perimeter. So, 100 - (30 + 40) = 30 cm. Therefore, the length of the third side is 30 cm.
10.
The height of an equilateral triangle of side 5 cm is:
Correct Answer
C. 4.33 cm
Explanation
The height of an equilateral triangle can be found using the formula h = (√3/2) * s, where h is the height and s is the length of a side. Plugging in the given side length of 5 cm into the formula, we get h = (√3/2) * 5 = 4.33 cm. Therefore, the correct answer is 4.33 cm.