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Why is AM perpendicular to BC in an isosceles triangle?

By Know.it editorial teamPublished Updated 5 min readEnglish

Why is AM perpendicular to BC in an isosceles triangle? — animated lesson thumbnail
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Quick answer

In triangle ABC with AB = AC, the line AM from the apex to the midpoint M of BC is perpendicular to BC. Triangles ABM and ACM have all three sides equal (AB = AC, BM = CM, AM common), so they're congruent. That makes angles AMB and AMC equal, and since they add to 180°, each is 90°.

In one lineIn an isosceles triangle, the line joining the apex to the midpoint of the base is perpendicular to the base, because it splits the triangle into two congruent right triangles.

Picture a perfectly balanced seesaw. Same weight on both ends, so it sits level. An isosceles triangle works the same way — two equal sides, and a hidden line inside that keeps everything in balance.

That line is AM, drawn from the top point A to the midpoint M of the base BC. Let's see why it always lands at exactly 90° to the base. No guessing needed — just two twin triangles.

What is an isosceles triangle?

An isosceles triangle is one with at least two sides of equal length. In our triangle ABC, side AB equals side AC. So A is the apex (the top point), and BC is the base.

Because the two sides are equal, the two angles opposite them are equal too. Angle ABC equals angle ACB. This is the isosceles triangle theorem, and it's the quiet reason everything else works.

M is the midpoint of BC, so BM = CM. That's all the setup we need.

Isosceles: two equal sides
Isosceles: two equal sides — Image: Wikimedia Commons

What does the line AM actually do?

Many people think AM is just a neat way to cut the triangle in half. But it does three precise jobs at once.

Here's the list to remember:

  • It splits the base BC exactly in half, so BM = CM.
  • It creates two smaller triangles, ABM and ACM, that are identical in every way.
  • It meets the base at exactly 90°, which means AM is perpendicular to BC.

Common mistake to avoid

Don't assume AM is perpendicular just because the triangle looks symmetric. In geometry, looks don't count — proof does. We have to show the 90° angle using sides and angles we already know. That's exactly what the next step does.

Why are triangles ABM and ACM congruent?

When we draw AM, we get two small triangles: ABM and ACM. Let's compare their sides one by one.

Side AB equals side AC — that's given. Side BM equals side CM — because M is the midpoint. And side AM is common to both triangles.

All three pairs of sides match. So by the SSS (side-side-side) congruence rule, triangle ABM is congruent to triangle ACM. They're perfect twins — same shape, same size, same angles.

Worked example with numbers

Say AB = AC = 5 cm and BC = 6 cm. Then BM = CM = 3 cm, and AM is shared. Sides are 5, 3, AM in both triangles. SSS holds, so the triangles are congruent. No measuring of angles needed — the sides already prove it.

Why must the angles at M be 90° each?

Now look at the two angles where AM meets BC: angle AMB and angle AMC.

Together they sit on the straight line BC, so they add up to 180°. That's the linear pair property.

Because the triangles are congruent, angle AMB equals angle AMC. Two equal angles adding to 180° means each one is 90°.

See? Simple. A 90° angle is the definition of perpendicular, so AM is perpendicular to BC. Done.

Can we prove it another way?

Yes, and it's a nice check. Since AB = AC, the base angles are equal: angle ABC = angle ACB.

Now compare triangles ABM and ACM using SAS: AB = AC, angle ABM = angle ACM, and BM = CM. Congruent again. So angle AMB = angle AMC, and the same 180° argument gives 90° each.

Both routes land in the same place. Pick whichever feels clearer to you in the exam.

Where does this show up in real life?

Think of a roof on a house. The two sloping sides are equal, and the vertical beam from the peak hits the base exactly in the middle — at 90°. That's AM in action.

The same idea appears in bridges, kite frames, and even the way a tent pole holds up the canvas. Symmetry plus a midpoint gives you a right angle, every time.

In exams, this result is often called the isosceles triangle property: the median from the apex is also the altitude and the angle bisector.

Key takeaways

  • In an isosceles triangle, the line from the apex to the midpoint of the base is perpendicular to the base.
  • Triangles ABM and ACM are congruent by SSS because AB = AC, BM = CM, and AM is common.
  • Congruent triangles give equal angles at M, and a linear pair adds to 180°, so each angle is 90°.
  • The same proof works with SAS using the equal base angles.
  • This line is the median, the altitude, and the angle bisector all at once.

Key terms

Isosceles triangle
A triangle with at least two sides of equal length.
Midpoint
The point that divides a line segment into two equal parts.
Perpendicular
Two lines meeting at exactly 90°.
Congruent triangles
Triangles with the same shape and size; all corresponding sides and angles are equal.
SSS rule
If three sides of one triangle equal three sides of another, the triangles are congruent.
Linear pair
Two adjacent angles on a straight line that add up to 180°.
Median
A line from a vertex to the midpoint of the opposite side.

Practice questions

  1. In triangle ABC with AB = AC and M the midpoint of BC, which pair of sides is equal because M is the midpoint?

    • (a) AB = AC
    • (b) BM = CM
    • (c) AM = AB
    • (d) AM = BC
    Answer key

    (b) BM = CM — A midpoint divides the segment into two equal parts, so BM = CM.

  2. Which congruence rule proves triangle ABM congruent to triangle ACM?

    • (a) AAA
    • (b) SSS
    • (c) RHS
    • (d) ASA
    Answer key

    (b) SSS — All three pairs of sides match: AB = AC, BM = CM, AM common.

  3. Angles AMB and AMC form a linear pair. What do they add up to?

    • (a) 90°
    • (b) 180°
    • (c) 270°
    • (d) 360°
    Answer key

    (b) 180° — Angles on a straight line always add to 180°.

  4. If angle AMB = angle AMC and they add to 180°, what is each angle?

    • (a) 45°
    • (b) 60°
    • (c) 90°
    • (d) 120°
    Answer key

    (c) 90° — Two equal angles adding to 180° must each be 90°.

  5. In an isosceles triangle, the median from the apex is also which of these?

    • (a) A side of the triangle
    • (b) The altitude and angle bisector
    • (c) The longest side
    • (d) Parallel to the base
    Answer key

    (b) The altitude and angle bisector — The proof shows AM is perpendicular to BC and bisects angle A.

Frequently asked questions

Why is AM perpendicular to BC in triangle ABC with AB = AC?

Because drawing AM creates two triangles, ABM and ACM, with all three sides equal. They're congruent by SSS, so angles AMB and AMC are equal. Since they form a straight line and add to 180°, each is 90°. A 90° angle means perpendicular.

Which congruence rule is used in this proof?

SSS is the main one: AB = AC (given), BM = CM (M is the midpoint), and AM is common. You can also use SAS, since the base angles of an isosceles triangle are equal.

Is AM also the angle bisector of angle A?

Yes. Because triangles ABM and ACM are congruent, angle BAM equals angle CAM. So AM bisects the apex angle. In an isosceles triangle, the median from the apex is also the altitude and the angle bisector.

What if the triangle is not isosceles?

Then the line from the apex to the midpoint of the base is not perpendicular to the base. The proof depends on AB = AC, so without equal sides, the result doesn't hold.

What does midpoint mean here?

M is the midpoint of BC, which means M lies on BC and BM = CM. It divides the base into two equal halves, and that equality is what makes the two small triangles congruent.

Quick recap: Draw AM from the apex to the midpoint of the base, and you get two triangles with all three sides equal. SSS makes them congruent, so the angles at M are equal — and since they add to 180°, each is 90°. That's why AM is perpendicular to BC.

Sources and further reading

Know.it editorial team

Written by the Know.it editorial team from the script of an animated Know.it lesson, checked against the lesson's own facts and the sources listed below. Know.it turns any question into a short visual lesson with voice and animation.

Reviewed: 11 November 2026 · Editorial standards · AI transparency

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