Grade 10 Math Similarity and Right Triangle Trigonometry quiz

📚 Quick Study Guide

• 📐 Similar Triangles: Triangles are similar if their corresponding angles are congruent and corresponding sides are in proportion. If $\triangle ABC \sim \triangle DEF$, then $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$.
• 💡 AA Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• 📏 SSS Similarity: If the corresponding sides of two triangles are proportional, then the triangles are similar.
• ✏️ SAS Similarity: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
• 🤓 Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. $a^2 + b^2 = c^2$.
• ✨ Trigonometric Ratios (SOH CAH TOA):
  • sin($\theta$) = $\frac{Opposite}{Hypotenuse}$
  • cos($\theta$) = $\frac{Adjacent}{Hypotenuse}$
  • tan($\theta$) = $\frac{Opposite}{Adjacent}$
• 🧭 Angles of Elevation and Depression: These are angles formed between a horizontal line and the line of sight to an object above (elevation) or below (depression).

Practice Quiz

1. In $\triangle ABC$, $AB = 6$, $BC = 8$, and $AC = 10$. In $\triangle DEF$, $DE = 3$, $EF = 4$, and $DF = 5$. Are the triangles similar?
   1. Yes
   2. No
   3. Cannot be determined
2. If $\triangle PQR \sim \triangle XYZ$, $\angle P = 50^{\circ}$, and $\angle Q = 70^{\circ}$, what is the measure of $\angle Z$?
   1. $50^{\circ}$
   2. $60^{\circ}$
   3. $70^{\circ}$
3. In a right triangle, if the opposite side is 3 and the hypotenuse is 5, what is the sine of the angle?
   1. 3/4
   2. 3/5
   3. 4/5
4. What trigonometric ratio is equal to $\frac{Adjacent}{Hypotenuse}$?
   1. Sine
   2. Cosine
   3. Tangent
5. A ladder leans against a wall, forming a $60^{\circ}$ angle with the ground. If the base of the ladder is 4 feet from the wall, how high up the wall does the ladder reach? (Use tangent)
   1. $4\sqrt{3}$ feet
   2. 4 feet
   3. $8$ feet
6. If sin($\theta$) = 0.6, what is cos($\theta$)? (Assume $\theta$ is an acute angle)
   1. 0.8
   2. 0.4
   3. 0.36
7. Two similar triangles have sides in the ratio 2:3. If the area of the smaller triangle is 20 $cm^2$, what is the area of the larger triangle?
   1. 30 $cm^2$
   2. 45 $cm^2$
   3. 60 $cm^2$