Construct & Bisect — The Triangle ZXY

STEP 1 · UNDERSTAND

Read the triangle first

Why start by reading it: a construction goes wrong when you grab the wrong length for the wrong side. So first work out which side is which — especially the longest one.

The right angle is at Y. The side opposite the right angle is the longest side — the hypotenuse — and that's XZ = 11 cm. The base XY = 8 cm sits beside the right angle.

The raw formula (Pythagoras)

a² + b² = c²

a, bthe two short sides that touch the right angle

cthe hypotenuse — always the side facing the right angle

We use it here to find the third side YZ, so later we can check our drawing is the right size.

↓ put XY = 8 and XZ = 11 in ↓

Fill in the numbers 8² + YZ² = 11² YZ² = 121 − 64 = 57 → YZ = √57 = 7.6 cm

Good to knowThird side YZ ≈ 7.6 cm

STEP 2 · CONSTRUCT

Lay the base & the right angle

Why this order: always build from the part you know for certain. We know the 8 cm base and that the corner at Y is exactly 90°, so we lock those down first.

The method

draw base → make 90° with a compass

With a ruler, draw XY = 8 cm. Mark X at one end, Y at the other.

Put the compass point on Y and draw a small arc that crosses the base on both sides of Y.

From those two crossings, swing two bigger arcs that meet above Y. The line from Y up through that meeting point is your exact 90° line.

Why a compass, not a protractor? A compass gives a perfect 90° every time. Protractors are easy to misread by a degree or two.

STEP 3 · CONSTRUCT

Find point Z, then join up

Why an arc: Z must be exactly 11 cm from X. Every point on a compass arc is the same distance from the centre — so an 11 cm arc from X marks every possible spot for Z at once. Where it crosses the 90° line is the only Z that also sits on it.

The method

compass = 11 cm, point on X, swing the arc

Open the compass to 11 cm against your ruler.

Place the point on X and draw an arc that cuts the upright 90° line.

That crossing is Z. Join X to Z with a ruler — the triangle is complete.

Answer (a)Triangle ZXY constructed

STEP 4 · BISECT

Split the corner at X in half

What "bisect" means: cut the angle ∠ZXY into two equal halves. We do it with a compass so both halves are guaranteed identical — no guessing.

The rule that makes it work

equal distances from both arms
→ the meeting point sits dead-centre

Keeping every compass opening the same forces perfect symmetry, so the line lands exactly down the middle.

Point the compass on X. Draw an arc that crosses both arms of the angle (mark the two crossings).

Without changing the width, put the point on each crossing and draw two arcs that meet inside the angle.

Draw a straight line from X through that meeting point. That line is the bisector.

DoneAngle ∠ZXY bisected

STEP 5 · CHECK

How to check it's accurate

Why check: the question literally asks you to explain how you'd know the bisector is correct. A good construction is one you can prove is right.

The check

measure both halves — they must be equal

Use a protractor: measure the whole angle ∠ZXY, then measure each half the bisector made.

What you should see whole angle ∠ZXY ≈ 43.3° each half ≈ 43.3 ÷ 2 = 21.7° → both halves equal = bisected correctly ✓

In words for your answer: "I measured angle ZXY with a protractor and then measured the two angles either side of the bisector. Because both halves are equal (about 21.7° each), the angle has been bisected accurately."

AnswerBoth halves equal → accurate

🎉 ALL DONE

Quick recap

Big idea: build from what you know for sure, and let the compass guarantee the exact angles a protractor might fumble.

1XZ = 11 cm is the hypotenuse; YZ ≈ 7.6 cm (Pythagoras)

2Draw the 8 cm base, then a compass-built 90° at Y

3Swing an 11 cm arc from X to find Z, then join

4Bisect ∠ZXY with equal compass arcs

5Check with a protractor: both halves ≈ 21.7°

Two power moves: ① an arc marks every point at the same distance from its centre. ② equal arcs from both sides of an angle always land the bisector dead-centre.