Problem restatement
We have an isosceles triangle with base = 24 and area = 60. Find the length of one of the equal (congruent) sides.
Step-by-step solution
- Use the area formula for a triangle: area = 1/2 × base × height.
So 60 = 1/2 × 24 × height. - Solve for the height: 60 = 12 × height, so height = 60 ÷ 12 = 5.
- Draw a perpendicular from the apex (where the equal sides meet) down to the base. In an isosceles triangle this perpendicular bisects the base, so each half of the base is 24 ÷ 2 = 12.
- Now consider the right triangle formed: one leg = height = 5, the other leg = half-base = 12, and the hypotenuse = one equal side s. Apply Pythagoras: s^2 = 12^2 + 5^2 = 144 + 25 = 169.
- So s = sqrt(169) = 13. The length of one equal side is 13 (units).
Evaluation of the student answer
The student wrote: "height is 5. half of base is 12. 144 and 25 is 169. answer is 13." This is correct. The student found the height from the area, recognised the half-base, applied Pythagoras, and obtained the correct final value 13. Good method and correct arithmetic. Minor improvement: explicitly show the area step (60 = 1/2 × 24 × h => h = 5) and include units.
Marking rubric (mapped to Year 8 geometry and measurement skills)
- Understanding (4 marks): Correctly uses area formula and recognises perpendicular bisects base. 4/4.
- Method and reasoning (4 marks): Steps are logically correct: find height, form right triangle, use Pythagoras. 4/4.
- Calculation accuracy (4 marks): Arithmetic correct (12^2 + 5^2 = 169). 4/4.
- Communication (4 marks): Clear but could be improved by stating formulas and units. 3/4.
Total: 15/16 (or full marks depending on rubric emphasis on communication).
Teacher comments (strict, direct tone)
You did exactly what I expect: you used the area to find the height, halved the base, then used Pythagoras to find the equal side. That is the right chain of reasoning. Your arithmetic is correct, and your final answer is correct. But let me be blunt: writing fragments is not enough. Mathematics is about clear arguments. You wrote steps, but you did not show the key formula application. Write the equation: 60 = 1/2 × 24 × h, then solve h = 5. State that the perpendicular bisects the base and why, even briefly. Always include units. These small habits — writing formulas, labeling steps, and including units — make your work easier to follow and make careless errors less likely. Now, push yourself. Try solving: if the equal side were 13 and base 24, confirm the area. Or change the base to 10 and the equal sides to 13 and find the area. Show full workings neatly next time. You are doing very well; tighten your presentation and you will be excellent.
Extension questions
- If the equal side is 13 and base = 24, what is the exact area? (Verify the original problem.)
- Given base = 24 and one equal side = 10, find the area.
- Generalise: for base b and area A of an isosceles triangle, express the equal side length in terms of b and A.