Right Triangles & Trigonometry 1 — Geometry and Trigonometry

Questions in this quiz (11)

1. In the right triangle shown, the hypotenuse has length $$13$$ and the base has length $$5$$. The angle at the left vertex is labeled $$\theta$$. What is the value of $$\cos\theta$$?

   • $$\frac{12}{13}$$
   • $$\frac{5}{13}$$
   • $$\frac{5}{12}$$
   • $$\frac{12}{5}$$

2. A right triangle has legs with lengths of $$8$$ meters and $$6$$ meters. If the length of this triangle's hypotenuse, in meters, can be written in the form $$k\sqrt{2}$$, where $$k$$ is an integer, what is the value of $$k$$?

   • $$5$$
   • $$6$$
   • $$8$$
   • $$12$$

3. An isosceles right triangle has a perimeter of $$60+60\sqrt{2}$$ centimeters. What is the length, in centimeters, of one leg of this triangle?

   • $$30$$
   • $$30\sqrt{2}$$
   • $$60$$
   • $$60\sqrt{2}$$

4. In a right triangle, the measures of the two acute angles are $$a^\circ$$ and $$b^\circ$$. If $$\sin(a^\circ)=\frac{5}{13}$$, what is the value of $$\cos(b^\circ)$$?

   • $$\frac{5}{13}$$
   • $$\frac{12}{13}$$
   • $$\frac{13}{12}$$
   • $$\frac{12}{5}$$

5. Triangles $$ABC$$ and $$DEF$$ are similar, with $$A$$ corresponding to $$D$$ and $$C$$ corresponding to $$F$$. Both $$\angle C$$ and $$\angle F$$ are right angles. Given that $$\tan(B)=\frac{1}{2}$$ and $$DE=60$$, what is the length of side $$EF$$?

   • $$30$$
   • $$24\sqrt{5}$$
   • $$30\sqrt{5}$$
   • $$60\sqrt{5}$$

6. The sine of an angle measuring $$(5x-10)^\circ$$ is equal to the cosine of an angle measuring $$(3x+20)^\circ$$. If both angles are acute, what is the value of $$x$$?

   • $$10$$
   • $$15$$
   • $$20$$
   • $$25$$

7. In triangle $$ABC$$, angle $$C$$ is a right angle. Point $$D$$ lies on line segment $$AB$$, point $$E$$ lies on line segment $$BC$$, and line segment $$DE$$ is parallel to $$AC$$. If the length of $$AC$$ is $$80$$ units, the length of $$DE$$ is $$20$$ units, and the area of triangle $$ABC$$ is $$1600$$ square units, what is the length of $$BE$$, in units?

   • $$5$$
   • $$8$$
   • $$10$$
   • $$12$$

8. In the figure above, $$\triangle PQR$$ is a right triangle. If $$\cos P=\frac{12}{13}$$, $$QR=15$$, and $$SQ=9$$, what is the length of $$ST$$?

   • $$\frac{5}{2}$$
   • $$5$$
   • $$8$$
   • $$\frac{45}{4}$$

9. The perimeter of an equilateral triangle is $$540$$ feet. The three vertices of the triangle lie on a circle. The radius of the circle is $$\frac{k}{\sqrt{3}}$$ feet. What is the value of $$k$$?

   • $$60$$
   • $$90$$
   • $$180$$
   • $$270$$

10. Which of the following expressions is equivalent to:$$\frac{\sin28^\circ}{\cos62^\circ}+\tan62^\circ$$

    • $$1$$
    • $$2$$
    • $$1+\cot28^\circ$$
    • $$1+\tan28^\circ$$

11. In the figure above, $$ABC$$ is a right triangle and $$2AC=3AB$$. If the quadrilateral $$AFED$$ is a square, then the area of the shaded region is what fraction of the area of triangle $$ABC$$?

    • $$\frac{3}{4}$$
    • $$\frac{2}{3}$$
    • $$\frac{13}{20}$$
    • $$\frac{12}{25}$$