Triangles ABC and DEF: angle A = angle D = 90, BC = EF = 41, AC = DF… Step by Step

Step by step

The question

Question: find x.

Triangles ABC and DEF have the marked parts equal.

Angle A = angle D = 90°.

BC = EF = 41.

AC = DF = 9.

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 A and angle D are both 90°: a pair of equal angles.

A for angle.

angle A = angle D = 90° A

BC and EF are both 41: a pair of equal sides.

S for side.

BC = EF = 41 S

AC and DF are both 9: a pair of equal sides.

S for side.

AC = DF = 9 S

2 Find the rule

Angle A is 90°: a right angle.

BC is across from it: the hypotenuse.

Right angle, hypotenuse, side: R H S.

angle A → BC → AC

Why?

Pythagoras: AB² is the hypotenuse squared take away the other side squared: 1600.

AB² = 41² − 9² = 1681 − 81 = 1600

So AB is 40.

DE is the same, 40.

All three sides match, like S S S.

AB = DE = 40

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

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

In congruent triangles, matching parts are equal.

So 3x + 1 = 40.

3x + 1 = 40

We want x by itself.

Two things are stuck to x: times 3 and + 1.

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

Then the times 3.

4 Remove the + 1

To remove + 1, we take away 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.

3x ~~+ 1 − 1~~ = 40 − 1

+ 1 and − 1 make zero.

Cross them out.

40 − 1 = 39.

40 − 1 = 39

Now we have 3x = 39.

3x = 39

5 Remove the 3

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

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

3x3 = 393

3 divided by 3 is 1.

So just x is left.

39 divided by 3 is 13.

393 = 13

So x = 13!

x = 13

6 Check

Let's check: put x = 13 in 3x + 1.

3x + 1 = 3 × 13 + 1 = 40

We get 40, the same as DE.

So x = 13 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 3.

Last: check.

Answer