Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Point $X$ is the midpoint of $AD$ and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. The ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form is $p:q$. Find $p+q$.

p+q = 12

Step-by-step explanation:

Let's draw the diagram obtained from the given information. Find attached the diagram.

When X and Y divides the line AD and line BY into half respectively, the diagram splits into two trapezoid.

See attachment for diagram.

Ratio of area ABYX to XYCD = p:q

p+q = ?

Area of trapezoid = ½(base 1 + base 2)height

For trapezoid ABYX:

Base 1 = AB = 12, Base 2 = XY

Using Pythagoras theorem to find height of ∆AOB

AO² = AP² + PO²

PO² = (10²-6²) = (100-36)

PO = √64 = 8

height of trapezoid ABYX =h = ½ PO

= 8÷2 = 4

base of triangles in trapezoid ABYX:

base² =  5² -4² = (25-16)

base = √9 = 3

XY = 12+3+3 = 18

Area of trapezoid ABYX = ½(AB + XY)height

= ½(12+18)×4

= ½(120) = 60

Area of trapezoid XYDC = ½(XY + DC)height

Height of both trapezoid = ½ PO = 4

base of triangles in trapezoid XYCD = 3

DC = 3+18+3 = 24

= ½(18+24)×4

= ½(168) = 84

Ratio of both areas

p:q = 60: 84

p:q = 5:7

p+q = 5+7

p+q = 12

c. 4 and 1/2 pounds is the answer! : )

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).