Solve a practical right-angled-triangle problem practice questions | Year 9 Mathematics

A practical right triangle

Use the physical endpoints and marked angle to interpret the sides. Check where each measurement begins and ends before choosing a ratio

Define the endpoints Decide what each side measures. A height above the observer is different from a height above the ground
Use the marked angle Opposite and adjacent depend on the stated angle. The hypotenuse is always opposite the right angle. An angle from the vertical can be used if the side roles match it
Choose from the known and wanted sides Use the ratios from the earlier topics. Keep all lengths in a common unit \(\displaystyle \begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\)
Interpret the answer Report the requested physical quantity. A hypotenuse must exceed either leg. Keep the full calculator value until the final rounding step

A straight brace supports a sign. Its horizontal run and angle from the horizontal are given. Find the brace length to one decimal place

\begin{aligned}a&=4.3\ \mathrm{m}\\\theta&=39^{\circ}\\L&>0\end{aligned}

\(\displaystyle \cos39^{\circ}=\frac{4.3}{L}\)The run is adjacent to the stated angle. The brace is the hypotenuse
\(\displaystyle L\cos39^{\circ}=L\left(\frac{4.3}{L}\right)\)Multiply both sides by the positive brace length
\(\displaystyle L\cos39^{\circ}=4.3\)Cancel the nonzero brace length on the right
\(\displaystyle \frac{L\cos39^{\circ}}{\cos39^{\circ}}=\frac{4.3}{\cos39^{\circ}}\)Divide both sides by the nonzero cosine
\(\displaystyle L=\frac{4.3}{\cos39^{\circ}}\)This expression gives the brace length
\(\displaystyle L\approx5.5330661\ \mathrm{m}\)Evaluate in degree mode and retain the full calculator value
\(\displaystyle L\approx5.5\ \mathrm{m}\)Round the brace length to one decimal place. It is longer than its horizontal run

Treating a ground height as a rise

✕\(\displaystyle o=H\)✓\(\displaystyle o=H-h_0\)

For an angle measured from the horizontal at the lower endpoint, the opposite side is the difference between the endpoint heights

Reversing a cosine ratio

✕\(\displaystyle \cos\theta=\frac{h}{a}\)✓\(\displaystyle \cos\theta=\frac{a}{h}\)

Cosine is adjacent divided by hypotenuse. The physical run is a leg when the angle is measured from the horizontal

Multiplying for an unknown denominator

✕\(\displaystyle h=a\cos\theta\)✓\(\displaystyle h=\frac{a}{\cos\theta}\)

Starting from adjacent divided by hypotenuse, multiply by the unknown hypotenuse and then divide by the nonzero cosine

introInterpret and calculate a numerator length using sine, cosine or tangent

standardCalculate a denominator length or use two sides to find the stated angle

Must every practical angle be measured from the horizontal?+

No. Use the reference the problem gives. If it uses a vertical line, name opposite and adjacent from that angle

When can I call the sloping side the hypotenuse?+

When the right angle is formed by a horizontal side and a vertical side. In any orientation, identify the hypotenuse by the right angle

Does a height always start at ground level?+

Read its endpoints. A vertical rise between two raised endpoints is their height difference, not the height of the upper endpoint alone

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 3: Pythagoras and trigonometry