MATHEMATICS MIDDLE SCHOOL

In rectangle $ABCD$, $P$ is a point on $BC$ so that $\angle APD=90^{\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. Find $SD$. (Express your answer as a common fraction.)

Answers

Answer 1

Answer:

Answer:

$(28)/(3)$

Step-by-step explanation:

Given information: ABCD is a rectangle, $\angle APD=90^(\circ)$ , PA=20, AQ=25 and QP=15.

In a right angled triangle

$hypotenuse^2=base^2+perpendicular^2$

In triangle ABP, AB = 16 and AP = 20. Using Pythagoras theorem we get

$(AB)^2 + (BP)^2 = (AP)^2$

$16^2 + (BP)^2 = 20^2$

$(BP)^2 = 20^2-16^2$

$BP^2 = 144$

$BP = 12$

Since BP = PT, therefore PT = 12.

$AS = BP + PT = 12 + 12 = 24$

AQS is a right angled triangle and AQ = 25. Use Pythagoras theorem in triangle AQS.

$(AS)^2 + (SQ)^2 = (AQ)^2$

$24^2 + (SQ)^2 = 25^2$

$SQ = 7$

Triangle PQT is a right triangle. Use Pythagoras theorem in triangle PQT.

$(PT)^2 + (QT)^2 = (PQ)^2$

$12^2 + (QT)^2 = 15^2$

$QT = 9$

In triangle PTQ and DSQ,

$\angle TQP=\angle SQD$ (Vertical angles)

$\angle PTQ=\angle DSQ$ (Right angles)

Triangle PTQ is similar to triangle DSQ by AA property of similarity.

Corresponding parts of similar triangles are proportional.

$(PT)/(QT)=(SD)/(SQ)$

$(12)/(9)=(SD)/(7)$

On cross multiplication we get

$9SD=12* 7$

$9SD=84$

Divide both sides by 9.

$SD=(84)/(9)$

$SD=(28)/(3)$

Therefore, $SD=(28)/(3)$ .

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9x–2(x+8) = 5x–11(2–x)

Find out X

Answers

Answer:

x = 6/9

Step-by-step explanation:

9x - 2(x+8) = 5x - 11(2-x)

9x - 2x - 16 = 5x -22 + 11x

collect the like terms

7x - 16 = 5x -22 + 11x

7x-5x = -22 +16 + 11x

2x = -6 + 11x

6 = 11x - 2x

6 = 9x

divide both sides by 9

6/9 = x

What is the quotient of 43,482 ÷ 12? with remainder.

Answers

the answer is 3623 with a remainder of 6

Answer:

3,623 R6

Step-by-step explanation:

A restaurant has square dining tables.they will place table settings on each table so that there is a distance of 2 feet from each corner to a table setting. The 3 table settings on each side will be 3 feet apart. What is the perimeter of one table?

Answers

Answer:

Perimeter = 12ft

Step-by-step explanation:

Given that the distance between the table and each corner of the table setting is 2feets it implies the table settings is place at the center of each table.

Since the table settings is at the center of the table and each table settings are 3feets apart, it implies that distance from table setting to the end of the table will be 3ft/2 = 1.5ft

If half length of the table is 1.5ft

Length = 2 x 1.5ft = 3ft

Formular for perimeter of a square = 4L

Perimeter = 4 x 3 = 12ft

See attachment for diagram.

9 OK there you go
if you don't think this is correct then ask someone else

To find 61 × 1,000, annex ______ zeros to _____ to form the product _______

Answers

The answer is 61,000

Final answer:

To multiply 61 by 1,000, you add 3 zeros to the end of 61 to get 61,000.

Explanation:

To find 61 × 1,000, you annex 3 zeros to 61 to form the product 61,000. In other words, when you multiply any number by 1,000, you simply add 3 zeros to the end of the original number. This is due to how the place values in our decimal number system work. Each position moving left in a number represents a ten-fold increase, so moving three places to the left (which is represented by multiplying by 1,000) corresponds to adding three zeros.