#1
In a right-angled triangle, what is the relationship between the lengths of the legs (a and b) and the hypotenuse (c) according to the Pythagorean theorem?
a^2 + b^2 = c^2
ExplanationThe sum of the squares of the legs is equal to the square of the hypotenuse.
#2
What are the angles of a 30-60-90 triangle?
30°, 60°, 90°
ExplanationAngles in a 30-60-90 triangle are 30°, 60°, and 90°.
#3
In a right-angled triangle, if one acute angle is 40 degrees, what is the measure of the other acute angle?
50 degrees
ExplanationThe sum of the angles in a right-angled triangle is 180 degrees; the other acute angle is 90 - 40 = 50 degrees.
#4
What is the name of the theorem that states the ratio of the lengths of the sides in a 45-45-90 triangle?
Isosceles triangle theorem
ExplanationThe Isosceles triangle theorem states the ratio of side lengths in a 45-45-90 triangle is 1:1:√2.
#5
If the angles of a triangle are in the ratio 1:2:3, what type of triangle is it?
Scalene
ExplanationThe unequal ratio of angles in a triangle indicates a Scalene triangle.
#6
In a 30-60-90 triangle, what is the ratio of the length of the longer leg to the length of the shorter leg?
1:√3
ExplanationThe ratio of the longer leg to the shorter leg in a 30-60-90 triangle is 1:√3.
#7
In a 45-45-90 triangle, what is the ratio of the lengths of the legs to the hypotenuse?
1:1:√2
ExplanationThe ratio of leg lengths to the hypotenuse in a 45-45-90 triangle is 1:1:√2.
#8
If the shorter leg of a 30-60-90 triangle is 4 units long, what is the length of the hypotenuse?
6
ExplanationIn a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg.
#9
If the hypotenuse of a right-angled triangle is 17 units long and one leg is 8 units long, what is the length of the other leg?
9 units
ExplanationUsing the Pythagorean theorem, the other leg length is calculated as √(hypotenuse^2 - leg length^2).
#10
What is the relationship between the angles in a 45-45-90 triangle?
Equal
ExplanationIn a 45-45-90 triangle, the two acute angles are equal.
#11
If the legs of a right-angled triangle are 5 units and 12 units, what is the length of the hypotenuse?
13 units
ExplanationUsing the Pythagorean theorem, the hypotenuse is calculated as √(leg1^2 + leg2^2).
#12
In a 45-45-90 triangle, if one leg is 10 units, what is the length of the other leg?
10√2 units
ExplanationUsing the Isosceles triangle theorem, the length of the other leg is the same, so it is 10√2 units.
#13
A ladder is leaning against a wall, forming a right-angled triangle. If the ladder is 10 meters long and the base (distance from the wall) is 8 meters, what is the height it reaches on the wall?
6 meters
ExplanationUsing the Pythagorean theorem, the height is calculated as √(ladder length^2 - base length^2).
#14
If the hypotenuse of a 30-60-90 triangle is 12 units long, what is the length of the shorter leg?
6 units
ExplanationIn a 30-60-90 triangle, the ratio of the shorter leg to the hypotenuse is 1:2; therefore, the shorter leg is half the length of the hypotenuse.
#15
A right-angled triangle has angles θ, 90°-θ, and 90°. If tan(θ) = 3/4, what is the value of cos(θ)?
4/5
ExplanationUsing trigonometric ratios, cos(θ) is calculated as adjacent/hypotenuse; here, cos(θ) = 4/5.
#16
A triangle has angles θ, 90°-θ, and 90°. If sin(θ) = 4/5, what is the value of cos(θ)?
3/5
ExplanationUsing trigonometric ratios, cos(θ) is calculated as adjacent/hypotenuse; here, cos(θ) = 3/5.
#17
A right-angled triangle has angles θ, 90°-θ, and 90°. If cos(θ) = 5/13, what is the value of tan(θ)?
5/12
ExplanationUsing trigonometric ratios, tan(θ) is calculated as opposite/adjacent; here, tan(θ) = 5/12.