Apply the Pythagorean theorem and the sine, cosine, and tangent ratios to find missing sides and angles in right triangles, including in real-world contexts such as angles of elevation.

What this topic is asking

The Geometry category requires you to solve right triangles (the G-SRT standards) using the Pythagorean theorem and the trigonometric ratios. On the Grade 10 MCAS you find missing sides and angles, including in contexts such as ladders, ramps, and angles of elevation. The Pythagorean theorem and the trig ratios are on the reference sheet, so the credit is for choosing the right tool and setting it up correctly.

The Pythagorean theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, always the longest):

Use it whenever you know two sides and want the third, and no angle is involved. To find the hypotenuse, add the squared legs and take the root; to find a leg, subtract: $a = \sqrt{c^2 - b^2}$.

Some side lengths recur as Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and their multiples (6-8-10, 9-12-15). Spotting these saves time, especially in the no-calculator session.

The trigonometric ratios

When an angle is involved, use a trig ratio. Relative to an acute angle $\theta$, label the sides: the hypotenuse (opposite the right angle), the side opposite $\theta$, and the side adjacent to $\theta$. Then:

To find a side given an angle and another side, pick the ratio that uses the known and unknown sides, then solve. To find an angle given two sides, use the inverse function: if $\tan\theta = \frac{3}{4}$, then $\theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 36.9^\circ$.

Choosing between the theorem and the ratios

The decision is quick once framed correctly:

• No angle, two sides known, third side wanted: use the Pythagorean theorem.
• An angle is given or wanted, and a side is involved: use a trig ratio (SOH-CAH-TOA), or its inverse to find an angle.

Angle-of-elevation and angle-of-depression problems are trig-ratio problems: the angle from the horizontal up to a line of sight (elevation) or down (depression) sits in a right triangle with the horizontal and vertical distances. A surveyor measuring the angle of elevation to the top of a tower, with the horizontal distance known, uses the tangent ratio to find the height, because the opposite (height) and adjacent (ground distance) are the relevant sides.

Special right triangles

Two right triangles have side ratios worth recognizing, and they appear on the reference sheet. The 45-45-90 triangle has legs in ratio $1 : 1 : \sqrt{2}$, so the hypotenuse is a leg times $\sqrt{2}$. The 30-60-90 triangle has sides in ratio $1 : \sqrt{3} : 2$, with the shortest side opposite the $30^\circ$ angle and the hypotenuse twice it. Knowing these lets you find exact side lengths without a calculator: a 45-45-90 triangle with legs 5 has hypotenuse $5\sqrt{2}$, and a 30-60-90 triangle with short side 4 has hypotenuse 8 and long leg $4\sqrt{3}$.

Try this

Q1. A right triangle has a leg of 9 and hypotenuse 15. Find the other leg.

• Cue. $\sqrt{15^2 - 9^2} = \sqrt{144} = 12$.

Q2. Which ratio finds a side from the angle and the adjacent side, wanting the opposite?

• Cue. Tangent (opposite over adjacent).