Triangles — Hard Practice Quiz

A Geometry cheat sheet for Triangles — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Pythagorean Theorem (Right Triangle)

$$a^2 + b^2 = c^2$$

Where: \(a, b\) = legs, \(c\) = hypotenuse

Area of Triangle

$$A = \frac{1}{2}bh$$

Where: \(b\) = base, \(h\) = height

Law of Sines

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

Where: \(R\) = circumradius

Law of Cosines

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Heron's Formula

$$A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}$$

Where: \(s\) = semi-perimeter

Equilateral Triangle

$$h = \frac{\sqrt{3}}{2}a, \quad A = \frac{\sqrt{3}}{4}a^2$$

Where: \(a\) = side length

Inradius and Circumradius

$$r = \frac{A}{s}, \quad R = \frac{abc}{4A}$$

Where: \(s\) = semi-perimeter, \(A\) = area

Median to Side a

$$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$

Altitude to Side a

$$h_a = \frac{2A}{a}$$

Practice quiz

  1. An equilateral triangle has a side length of $a$. If its inradius is $r$ and its circumradius is $R$, what is the ratio $\frac{R}{r}$?

    • $1$
    • $2$
    • $\frac{1}{2}$
    • $\sqrt{3}$

    Answer: $2$

  2. A triangle has sides $a=7$, $b=8$, and the angle $C$ opposite side $c$ is $60^\circ$. What is the length of the altitude to side $a$?

    • $2\sqrt{3}$
    • $4\sqrt{3}$
    • $7\sqrt{3}$
    • $8\sqrt{3}$

    Answer: $4\sqrt{3}$

  3. In a triangle $ABC$, $A=30^\circ$, $B=45^\circ$, and side $a=10$. What is the area of the triangle?

    • $25(\sqrt{3}+1)$
    • $50(\sqrt{3}+1)$
    • $25\sqrt{3}$
    • $50\sqrt{2}$

    Answer: $25(\sqrt{3}+1)$

  4. In a triangle $ABC$, if side $b$ is doubled while sides $a$ and $c$ remain constant, how does the length of the median to side $a$, $m_a$, change?

    • It doubles.
    • It quadruples.
    • It increases, but not necessarily by a simple factor.
    • It decreases.

    Answer: It increases, but not necessarily by a simple factor.

  5. A right-angled triangle has legs of length $6$ and $8$. What is the sum of its inradius and circumradius?

    • $5$
    • $7$
    • $10$
    • $12$

    Answer: $7$

  6. A triangle has sides $a=13$, $b=14$, $c=15$. What is the value of $\sin A$?

    • $\frac{3}{5}$
    • $\frac{4}{5}$
    • $\frac{12}{13}$
    • $\frac{13}{14}$

    Answer: $\frac{4}{5}$

  7. In a triangle $ABC$, if side $a$ is doubled while sides $b$ and $c$ remain constant, which of the following statements is true regarding the angles and the Law of Sines?

    • Angle $A$ decreases, and the ratio $\frac{a}{\sin A}$ remains constant.
    • Angle $A$ increases, and the ratio $\frac{a}{\sin A}$ doubles.
    • Angle $A$ increases, and the ratio $\frac{a}{\sin A}$ increases, but not necessarily doubles.
    • Angle $A$ remains constant, and the ratio $\frac{a}{\sin A}$ doubles.

    Answer: Angle $A$ increases, and the ratio $\frac{a}{\sin A}$ increases, but not necessarily doubles.

  8. If the side length of an equilateral triangle is increased by $50\%$, by what percentage does its area increase?

    • $50\%$
    • $100\%$
    • $125\%$
    • $225\%$

    Answer: $125\%$

  9. In a triangle $ABC$, the altitude to side $a$ is $h_a=12$. If $b=13$ and $c=15$, what is the length of side $a$?

    • $4$
    • $10$
    • $14$
    • $16$

    Answer: $14$

  10. In a triangle $ABC$, if $m_a$ is the median to side $a$, and $h_a$ is the altitude to side $a$, which of the following statements is always true?

    • $m_a = h_a$ if $b=c$.
    • $m_a < h_a$.
    • $m_a = \frac{1}{2}a$.
    • $m_a^2 + h_a^2 = b^2+c^2$.

    Answer: $m_a = h_a$ if $b=c$.

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