Right-triangle trig: Geometry Regents Practice

In right triangle $DEF$, the right angle is at $F$. The hypotenuse $DE = 20$ and $m\angle D = 60^\circ$. What is the length of $DF$, the side adjacent to $\angle D$?

• (A) $10$ ✓
• (B) $40$
• (C) $34.6$
• (D) $17.3$

Choose the ratio that links the angle, the hypotenuse and the unknown adjacent side; adjacent over hypotenuse is cosine.

In a right triangle, the side opposite acute angle $A$ has length $3$ and the hypotenuse has length $5$. What is $m\angle A$, to the nearest degree?

• (A) $53^\circ$
• (B) $31^\circ$
• (C) $37^\circ$ ✓
• (D) $60^\circ$

When a side and the hypotenuse are known, apply inverse sine to recover the angle.

A 20-foot ladder leans against a vertical wall so that the ladder makes an angle of $72^\circ$ with the level ground. Determine, to the nearest tenth of a foot, how far up the wall the top of the ladder reaches.

The ladder, wall, and ground form a right triangle in which the ladder is the hypotenuse. The height on the wall is opposite the $72^\circ$ angle, so the sine ratio applies: $h = 20\sin 72^\circ \approx 19.0$ feet.

A 20-foot ladder leans against a vertical wall so that it makes an angle of $65°$ with the level ground. To the nearest tenth of a foot, how high up the wall does the ladder reach?

• (A) $22.1$
• (B) $8.5$
• (C) $18.1$ ✓
• (D) $42.9$

The wall height is the side opposite the $65°$ angle and the ladder is the hypotenuse, so $h = 20\sin 65° \approx 18.1$ feet. Using cosine gives $8.5$ (the distance along the ground), tangent gives $42.9$, and dividing by sine gives $22.1$, all common setup errors.

A $20$-foot ladder leans against a vertical wall, making an angle of $70^\circ$ with the level ground. To the nearest tenth of a foot, how high up the wall does the ladder reach?

• (A) $21.3$
• (B) $6.8$
• (C) $54.9$
• (D) $18.8$ ✓

The ladder is the hypotenuse of a right triangle, and the height on the wall is the side opposite the $70^\circ$ angle, so $h = 20\sin 70^\circ \approx 18.8$ feet. Using cosine instead gives $6.8$, the distance of the foot of the ladder from the wall, a common mix-up.

A surveyor stands at a point on level ground $40$ feet from the base of a tree. From that point, the angle of elevation to the top of the tree measures $52°$. Determine and state, to the nearest foot, the height of the tree.

The tree, the ground, and the line of sight form a right triangle. The tangent ratio relates the angle of elevation to the opposite side (the tree's height) and the adjacent side (the horizontal distance), giving $h = 40\tan 52° \approx 51$ feet.

A 20-foot ladder leans against a vertical wall so that it makes an angle of $72^\circ$ with the level ground. Determine and state, to the nearest tenth of a foot, how far up the wall the ladder reaches.

The ladder, wall, and ground form a right triangle where the ladder is the hypotenuse. The height on the wall is opposite the $72^\circ$ angle, so the sine ratio applies: $h = 20\sin 72^\circ \approx 19.0$ feet.

A surveyor stands at a point on level ground that is $180$ feet from the base of a vertical cell phone tower. From that point, the angle of elevation to the top of the tower is $52°$. Determine and state, to the nearest foot, the height of the tower.

The tower, the ground, and the line of sight form a right triangle. Since the height of the tower is the side opposite the $52°$ angle and the $180$-foot distance is the adjacent side, the tangent ratio applies: $h = 180\tan 52° \approx 230$ feet.

In right triangle $ABC$, the right angle is at $C$. If $AB = 10$ and the angle at $A$ is $30^\circ$, what is the length of $BC$ (the side opposite $A$)?

• (A) $5$ ✓
• (B) $8.7$
• (C) $10$
• (D) $20$

Pick the trig ratio that links the angle, the known side and the unknown side. Opposite over hypotenuse is sine.

In right triangle $ABC$, the right angle is at $C$, $AB=10$, and $BC=6$. What is $\sin A$?

• (A) $\tfrac{6}{8}$
• (B) $\tfrac{10}{6}$
• (C) $\tfrac{8}{10}$
• (D) $\tfrac{6}{10}$ ✓

By SOH-CAH-TOA, sin A is the side opposite angle A (BC) over the hypotenuse (AB).

A $12$-foot ladder leans against a wall, making a $65^\circ$ angle with the level ground. How high up the wall does the top of the ladder reach, to the nearest tenth of a foot?

• (A) $25.7$
• (B) $13.2$
• (C) $10.9$ ✓
• (D) $5.1$

The ladder is the hypotenuse and the wall height is the opposite side, so the height is hypotenuse times sine of the angle.

From the top of a $50$-foot lighthouse, the angle of depression to a boat is $25^\circ$. How far is the boat from the base of the lighthouse, to the nearest foot?

• (A) $107$ ✓
• (B) $55$
• (C) $23$
• (D) $118$

With the opposite side known and the adjacent side unknown, use tangent and solve by dividing.

In right triangle $ABC$, the right angle is at $C$, $AC = 12$, and $BC = 5$. What is the value of $\sin A$?

• (A) $\frac{5}{12}$
• (B) $\frac{13}{5}$
• (C) $\frac{12}{13}$
• (D) $\frac{5}{13}$ ✓

Sine is opposite over hypotenuse; the side opposite $A$ is $BC = 5$ and the hypotenuse is 13.

A 20-foot ladder leans against a vertical wall so that the ladder makes an angle of $70°$ with the level ground. How high up the wall, to the nearest tenth of a foot, does the ladder reach?

• (A) $18.8$ ✓
• (B) $21.3$
• (C) $54.9$
• (D) $6.8$

The ladder is the hypotenuse of a right triangle, and the height reached is opposite the 70 degree angle, so h = 20 sin 70 ≈ 18.8 feet. Using cosine gives 6.8 (the distance from the wall) and using tangent gives 54.9, both common errors.

A surveyor stands at a point on level ground that is $80$ feet from the base of a building. From that point, the angle of elevation to the top of the building is $34°$, and the angle of elevation to the top of an antenna that sits on top of the building is $40°$. Determine and state, to the nearest foot, the height of the antenna.

Both the building top and the antenna top lie above the same base point, so each height is the opposite leg of a right triangle with the same 80-foot adjacent leg. Using the tangent ratio twice and subtracting gives the height of just the antenna, about 13 feet.

If $\sin(2x + 10)^\circ = \cos(3x - 15)^\circ$, where $2x + 10$ and $3x - 15$ are the measures of acute angles, what is the value of $x$?

• (A) $19$ ✓
• (B) $17$
• (C) $5$
• (D) $25$

Sine and cosine are cofunctions: $\sin\theta = \cos(90^\circ - \theta)$, so the two acute angles must sum to $90^\circ$. Solving $(2x+10)+(3x-15)=90$ gives $x = 19$. Setting the angle expressions equal to each other, a common error, gives $25$.

A 20-foot ladder leans against a vertical wall so that the ladder makes an angle of $72^\circ$ with the level ground. Determine and state, to the nearest tenth of a foot, how high up the wall the ladder reaches.

The ladder, wall, and ground form a right triangle in which the ladder is the hypotenuse and the height on the wall is opposite the $72^\circ$ angle. Using sine, the height is $20\sin 72^\circ \approx 19.0$ feet.

A $20$-foot ladder leans against a vertical wall so that the ladder makes an angle of $72°$ with the level ground. How high up the wall, to the nearest tenth of a foot, does the ladder reach?

• (A) $61.6$
• (B) $6.2$
• (C) $19.0$ ✓
• (D) $21.0$

The height on the wall is the side opposite the 72 degree angle, and the 20-foot ladder is the hypotenuse, so sine is the correct ratio: h = 20 sin 72, about 19.0 feet. Using cosine gives 6.2 and using tangent gives 61.6, the common setup errors.

In right triangle $ABC$, the right angle is at $C$, $m\angle A = 32^{\circ}$, and hypotenuse $AB = 20$. What is the length of $\overline{BC}$, to the nearest tenth?

• (A) $12.5$
• (B) $23.6$
• (C) $17.0$
• (D) $10.6$ ✓

Since the right angle is at $C$, side $BC$ is opposite the $32^{\circ}$ angle at $A$, and $AB = 20$ is the hypotenuse. The sine ratio gives $BC = 20\sin 32^{\circ} \approx 10.6$. Using cosine instead gives the adjacent side $17.0$, and using tangent gives $12.5$, both common errors from misidentifying the side relative to the angle.

A 20-foot ladder leans against a vertical wall so that the ladder makes an angle of $68^\circ$ with the level ground. Determine and state, to the nearest tenth of a foot, how high up the wall the ladder reaches.

The ladder, wall, and ground form a right triangle in which the ladder is the hypotenuse and the height on the wall is the leg opposite the $68^\circ$ angle. Since sine relates the opposite leg to the hypotenuse, $h = 20\sin 68^\circ \approx 18.5$ feet.

A $20$-foot ladder leans against a vertical wall, making an angle of $70°$ with the level ground. How high up the wall, to the nearest tenth of a foot, does the ladder reach?

• (A) $6.8$
• (B) $54.9$
• (C) $21.3$
• (D) $18.8$ ✓

The ladder is the hypotenuse of a right triangle and the height on the wall is opposite the 70 degree angle, so sine applies. Using cosine gives 6.8, tangent gives 54.9, and dividing instead of multiplying gives 21.3, all common errors.

A 20-foot ladder leans against a vertical wall so that the ladder makes an angle of $70°$ with the level ground. To the nearest tenth of a foot, how far up the wall does the ladder reach?

• (A) $18.8$ ✓
• (B) $6.8$
• (C) $54.9$
• (D) $7.3$

The ladder, wall, and ground form a right triangle in which the ladder is the hypotenuse and the height on the wall is the side opposite the $70°$ angle. Therefore $h = 20\sin 70° \approx 18.8$ feet. Using cosine gives $6.8$ (the distance along the ground), and using tangent gives $54.9$, both common errors.

In right triangle $ABC$ (right angle at $C$), the two acute angles are $A$ and $B$. Which statement is always true?

• (A) $\tan A = \tan B$
• (B) $\sin A = \cos B$ ✓
• (C) $\cos A = \cos B$
• (D) $\sin A = \sin B$

The acute angles are complementary, and the sine of an angle equals the cosine of its complement.

A ramp rises at an angle of $15^\circ$ and covers a horizontal distance of $20$ feet. Find the height $h$ of the ramp to the nearest tenth of a foot.

The height is opposite the 15° angle and 20 ft is adjacent, so use tangent and solve h = 20·tan 15°.