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Mathematics

1 answer:

Pie3 years ago

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Answer:

Step-by-step explanation:

All these amount of money are based on Dan's initial amount, which is x. Ben had 7 more than Dan, so Ben's amount of money is x + 7. Kaden has 4 more than Ben, so Kaden's amount of money is 4 + (x + 7) which is x + 11. So here's the breakdown so far:

Dan: x

Ben: x + 7

Kaden: x + 11

Now it looks like Kaden is the favorite and got another 20 from his mom, so his amount became x + 31. After Dan spends a dollar, making his amount x - 1, Kaden had 9 times what Dan had. Now we have finally:

x + 31 = 9(x - 1) and

x + 31 = 9x - 9 and

40 = 8x so

x = 5.

Dan originally had $5, Ben originally had $12, and Kaden originally had $16

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ABCD is a rectangle. The measure of angle BCE is 4x-23 and the angle measure of angle DCE is 5+5x. Find the measure of angle BEC

Temka [501]

<h2>Explanation:</h2><h2></h2>

The diagram for this problem is shown below. Point E is the intersection of the two diagonals and we know that:

$\angle BCE=\beta \\ \\ \angle DCE=\theta$

Every internal angle of any rectangle measures 90 degrees, so:

$\beta +\alpha=90 \\ \\ (4x-23)+(5+5x)=90 \\ \\ \\ Isolating \ x: \\ \\ (4x+5x)+(5-23)=90 \\ \\ 9x-18=90 \\ \\ 9x=108 \\ \\ x=\frac{108}{9}=12$

So:

$\beta=4x-23=4(12)-23=25^{\circ}$

So the measure of angle BEC can be found as follows:

We know that triangle ΔCEB is an isosceles triangle because the diagonals of any rectangle measure the same. So

$\angle EBC=\beta$

The sum of internal angles of any triangle add up to 180 degrees, so:

$\beta + \beta+\angle BEC=180^{\circ} \\ \\ 2\beta+\angle BEC=180^{\circ} \\ \\ \angle BEC=180^{\circ}-2\beta \\ \\ \angle BEC=180^{\circ}-2(25^{\circ}) \\ \\ \angle BEC=180^{\circ}-50^{\circ} \\ \\ \boxed{\angle BEC=130^{\circ}}}$