KJSEA 2025 Maths Q31 — Pythagoras & Density

KJSEA 2025 Grade 9 Measurement

Published

The Question

“Juma placed a metal rod leaning against a wall. The foot of the rod was 3.6 m from the base of the wall, and the top of the rod touched the wall 4.8 m above the ground, forming a right-angled triangle. (a) Using Pythagoras' theorem, find the length of the rod. (b) A metal object has a mass of 2.4 kg and a volume of 600 cubic centimetres. Find its density in grams per cubic centimetre.”

1

Picture the right-angled triangle

The wall stands straight up and the ground is flat, so where they meet there is a right angle. The rod leans from the ground to the wall, joining the two ends. That means the two short sides are the distance along the ground and the height up the wall, while the rod itself is the hypotenuse, the longest side opposite the right angle.

ground=3.6 m,height=4.8 m\text{ground} = 3.6\ \text{m}, \quad \text{height} = 4.8\ \text{m}
2

Apply Pythagoras' theorem

Pythagoras' theorem says the square of the hypotenuse equals the sum of the squares of the other two sides. Square each short side and add them to get the square of the rod's length.

rod2=3.62+4.82\text{rod}^{2} = 3.6^{2} + 4.8^{2}
rod2=12.96+23.04=36\text{rod}^{2} = 12.96 + 23.04 = 36
3

Take the square root to find the rod

The rod squared is 36, so the rod itself is the square root of 36. Since a length must be positive, we take the positive root.

rod=36=6 m\text{rod} = \sqrt{36} = 6\ \text{m}
4

Set up density for part (b)

Density is how much mass is packed into each unit of volume, found by dividing mass by volume. Because the answer is asked for in grams per cubic centimetre, we must first change the mass from kilograms into grams so the units match.

density=massvolume\text{density} = \frac{\text{mass}}{\text{volume}}
2.4 kg=2.4×1000=2400 g2.4\ \text{kg} = 2.4 \times 1000 = 2400\ \text{g}
5

Divide mass by volume

Now divide the mass in grams by the volume in cubic centimetres to get the density in grams per cubic centimetre.

density=2400 g600 cm3=4 g/cm3\text{density} = \frac{2400\ \text{g}}{600\ \text{cm}^{3}} = 4\ \text{g/cm}^{3}

Final Result

The rod is 6 m long, and the density of the metal object is 4 grams per cubic centimetre.

Why this method works

Pythagoras' theorem works because the wall meets the ground at a right angle, and in any right-angled triangle the area of the square on the longest side exactly equals the combined areas of the squares on the two shorter sides; that fixed relationship lets us recover the missing side from the two we know. Density works as a division because it measures concentration of matter — how much mass sits in each cubic centimetre — so sharing the total mass equally over every unit of volume gives the value per unit. Converting kilograms to grams before dividing is essential so the units of the answer come out as grams per cubic centimetre.

✓

Reverse-check part (a): 6² = 36 and 3.6² + 4.8² = 12.96 + 23.04 = 36, so they match. Part (b): 4 g/cm³ × 600 cm³ = 2400 g = 2.4 kg, the original mass.