Answer:
16) The area of the circle is 25.1 units²
17) JKLM is a parallelogram but not a rectangle
Explanation:
16) Lets talk about the area of the circle
- To find the area of the circle you must find the length of the radius
- In the problem you have the center of the circle and a point on
the circle, so you can find the length of the radius by using the
distance rule
* Lets solve the problem
∵ The center of the circle is (1 , 3)
∵ The point on the circle is (3 , 5)
- Using the rule of the distance between two points
* Lets revise it
- The distance between the two points (x1 , y1) and (x2 , y2) is:
Distance = √[(x2 - x1)² + (y2 - y1)²]
∴ r = √[(3 - 1)² + (5 - 3)²] = √[4 + 4] = √8 = 2√2 units
∵ The area of the circle = πr²
∴ The area of the circle = π (2√2)² = 8π = 25.1 units²
* The area of the circle is 25.1 units²
17) To prove a quadrilateral is a parallelogram, prove that every
to sides are parallel or equal or the two diagonal bisect
each other
* The parallelogram can be rectangle if two adjacent sides are
perpendicular to each other (measure of angle between them is 90°)
or its diagonals are equal in length
- The parallel lines have equal slopes, then to prove the
quadrilateral is a parallelogram, we will find the slopes of
each opposite sides
* Lets find from the graph the vertices of the quadrilateral
∵ J = (0 ,2) , K (2 , 5) , L (5 , 0) , M (3 , -3)
- The opposite sides are JK , ML and JM , KL
- The slope of any line passing through point (x1 , y1) and (x2 , y2) is
m = (y2 - y1)/(x2 - x1)
∵ The slop of JK = (5 - 2)/(2 - 0) = 3/2 ⇒ (1)
∵ The slope of LM = (-3 - 0)/(3 - 5)= -3/-2 = 3/2 ⇒ (2)
- From (1) and (2)
∴ JK // LM
∵ The slope of KL = (0 - 5)/(5 - 2) = -5/3 ⇒ (3)
∵ The slope of JM = (-3 - 2)/(3 - 0)= -5/3 ⇒ (4)
- From (3) and (4)
∴ KL // JM
∵ Each two opposite sides are parallel in the quadrilateral JKLM
∴ It is a parallelogram
- The product of the slopes of the perpendicular line is -1
* lets check the slopes of two adjacent sides in the JKLM
∵ The slope of JK = 3/2 and the slope of KL = -5/3
∵ 3/2 × -5/3 = -5/2 ≠ -1
∴ JKLM is a parallelogram but not a rectangle