Math  /  Geometry

QuestionABCDEF;mA=\triangle A B C-\triangle D E F ; m \angle A= \qquad ; m<E=\mathrm{m}<\mathrm{E}= \qquad x=x= \qquad ; y= The perimeter of ABC is 36.\begin{array}{l} y= \\ \text { The perimeter of } \triangle A B C \text { is } 36 . \end{array}

Studdy Solution

STEP 1

What is this asking? We've got two similar triangles, and we need to find the lengths of two sides of one triangle, knowing the lengths of all sides of the other triangle, and the perimeter of the triangle we're solving for. Watch out! Make sure you match corresponding angles and sides correctly between the similar triangles!

STEP 2

1. Find the missing angle
2. Match corresponding sides
3. Set up proportions
4. Solve for x
5. Solve for y

STEP 3

Alright, first things first!
We know two angles in DEF\triangle DEF, E=82\angle E = 82^\circ and F=34\angle F = 34^\circ.
Since the angles in a triangle add up to 180180^\circ, we can find D\angle D.

STEP 4

So, D+E+F=180\angle D + \angle E + \angle F = 180^\circ.
Plugging in the values we know, we get D+82+34=180\angle D + 82^\circ + 34^\circ = 180^\circ.

STEP 5

Simplifying, we have D+116=180\angle D + 116^\circ = 180^\circ.
Subtracting 116116^\circ from both sides gives us D=180116=64\angle D = 180^\circ - 116^\circ = 64^\circ.
Boom!

STEP 6

We're told ABC\triangle ABC and DEF\triangle DEF are similar.
We know C\angle C in ABC\triangle ABC is 3434^\circ, which matches F\angle F in DEF\triangle DEF.
Also, we just found that D=64\angle D = 64^\circ, and since we don't have any other info, we can assume A=D=64\angle A = \angle D = 64^\circ.
This means B\angle B must be 8282^\circ to make the angles in ABC\triangle ABC add up to 180180^\circ.

STEP 7

So, we have A\angle A matching D\angle D, B\angle B matching E\angle E, and C\angle C matching F\angle F.
This tells us which sides correspond: ABAB with DEDE, BCBC with EFEF, and ACAC with DFDF.

STEP 8

Since the triangles are similar, the ratios of corresponding sides are equal.
We can set up the proportion: ABDE=BCEF=ACDF \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}

STEP 9

Substituting the given values, we get: x7=y12=z14 \frac{x}{7} = \frac{y}{12} = \frac{z}{14}

STEP 10

We know the perimeter of ABC\triangle ABC is 3636, which means x+y+z=36x + y + z = 36.

STEP 11

Let's express yy and zz in terms of xx using our proportions.
From x7=y12\frac{x}{7} = \frac{y}{12}, we get y=12x7y = \frac{12x}{7}.
From x7=z14\frac{x}{7} = \frac{z}{14}, we get z=14x7=2xz = \frac{14x}{7} = 2x.

STEP 12

Now substitute these into the perimeter equation: x+12x7+2x=36x + \frac{12x}{7} + 2x = 36.

STEP 13

To solve for xx, combine the xx terms: 3x+12x7=363x + \frac{12x}{7} = 36, which is 21x7+12x7=33x7=36\frac{21x}{7} + \frac{12x}{7} = \frac{33x}{7} = 36.

STEP 14

Multiply both sides by 77 to get 33x=25233x = 252.
Then, divide by 3333 to find x=25233=8411x = \frac{252}{33} = \frac{84}{11}.

STEP 15

We already found that y=12x7y = \frac{12x}{7}.
Now we just plug in our value for xx: y=1278411y = \frac{12}{7} \cdot \frac{84}{11}.

STEP 16

Simplifying, we get y=121211=14411y = \frac{12 \cdot 12}{11} = \frac{144}{11}.

STEP 17

x=8411x = \frac{84}{11} and y=14411y = \frac{144}{11}.

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