Triangles ABC and DEF: angle B = angle E = 90, AC = DF = 13, BC = EF… Step by Step

Step by step

The question

Question: find x.

Triangles ABC and DEF have the marked parts equal.

Angle B = angle E = 90°.

AC = DF = 13.

BC = EF = 12.

1 Match the parts

Congruent means the same shape and the same size.

One fits exactly on top of the other.

The order of the letters shows which corners match: A with D, B with E, C with F.

DEF is turned over.

That does not matter.

Angle B and angle E are both 90°: a pair of equal angles.

A for angle.

angle B = angle E = 90° A

AC and DF are both 13: a pair of equal sides.

S for side.

AC = DF = 13 S

BC and EF are both 12: a pair of equal sides.

S for side.

BC = EF = 12 S

2 Find the rule

Angle B is 90°: a right angle.

AC is across from it: the hypotenuse.

Right angle, hypotenuse, side: R H S.

angle B → AC → BC

Why?

Pythagoras: AB² is the hypotenuse squared take away the other side squared: 25.

AB² = 13² − 12² = 169 − 144 = 25

So AB is 5.

DE is the same, 5.

All three sides match, like S S S.

AB = DE = 5

Careful!

S S A is not a rule.

But with a right angle and the hypotenuse, it is: R H S.

Careful: S S A is not a rule, but R H S is

Pythagoras fixes the third side, so all three sides match.

There is only one way to draw it.

So R H S shows the triangles are congruent.

It is also called H L: hypotenuse, leg.

RHS → congruent

3 Use the matching parts

DE matches AB: D with A, E with B.

In congruent triangles, matching parts are equal.

So 2x − 1 = 5.

2x − 1 = 5

We want x by itself.

Two things are stuck to x: times 2 and − 1.

The − 1 is on the outside, so it comes off first.

Then the times 2.

4 Remove the − 1

To remove − 1, we add 1.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

2x ~~− 1 + 1~~ = 5 + 1

− 1 and + 1 make zero.

Cross them out.

5 + 1 = 6.

5 + 1 = 6

Now we have 2x = 6.

2x = 6

5 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2x2 = 62

2 divided by 2 is 1.

So just x is left.

6 divided by 2 is 3.

62 = 3

So x = 3!

x = 3

6 Check

Let's check: put x = 3 in 2x − 1.

2x − 1 = 2 × 3 − 1 = 5

We get 5, the same as AB.

So x = 3 is right!

Let's look back.

First: match the parts.

Then: find the rule.

Then: use the matching parts.

Then: remove the − 1.

Then: remove the 2.

Last: check.

Answer