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This investigation looks at the area of a rhombus drawn on a square grid.
The first square has area 1.
Find the area of each square in the diagram. Write your answer inside each square.
585
578
540
Area of a triangle = \( \frac{1}{2} \times \text{base} \times \text{height} \)
(a) ![Image_1 of a triangle with area 5]
Show that the area of this triangle is 5. [1]
(b) ![Image_2 of two triangles]
Find the area of each triangle.
Write your answer inside each triangle. [3]
558
571
586
(a) Give a reason why $OPQR$ is a rhombus.
................................................................................................. [1]
(b) These steps are the start of a method to find the area of the rhombus $OPQR$.
$\textbf{Step 1}$ Draw a square around the rhombus.
$\textbf{Step 2}$ Fill the space between the square and the rhombus with two congruent squares and four congruent triangles.
(i) Find the area of the large square that goes around the rhombus.
....................................................... [2]
(ii) Use some of the results in $\text{Question 1}$ and $\text{Question 2}$ to write the areas of the two congruent squares and the four congruent triangles inside each shape. [1]
(iii) Use your answers to $\text{part (i)}$ and $\text{part (ii)}$ to show that the area of the rhombus is 21. [1]
567
569
569
The diagram shows another rhombus $OPQR$.
Use the method of Question 3(b) to find its area.
594
546
534
(a) On the diagram, complete the rhombus OPQR. [1]
(b) Use the method of Question 3(b) to find the area of rhombus OPQR that you completed in part (a). [4]
570
589
532
Throughout this investigation, $O$ is the origin, and the $x$-coordinate and the $y$-coordinate of $Q$ are always equal.
Complete the table using your answers to Question 4 and Question 5 and any patterns you notice.
[Table_1]
Area of rhombus $OPQR$ | $P$ | $Q$
21 | $(5, 2)$ | $(7, 7)$ | $5^2 - 2^2 = 21$
Question 4 | $(3, 2)$ | $(5, 5)$ | $3^2 - 2^2 =$
Question 5 | $(4, 1)$ | $(4, 1)$ | $4^2 - 1^2 =$
$(7, 4)$ | $(11, 11)$ | $= 33$
56 | $(9, 5)$ |
27 | $(9, 9)$ | $6^2 - 3^2 =$
527
519
558
OPQR is a rhombus with O(0, 0) and P(a, b) where a > b.
(a) Use the table in Question 6 to
(i) write down the coordinates of Q in terms of a and b
( ......................... , .........................) [1]
(ii) write an expression for the area of rhombus OPQR in terms of a and b.
.................................................... [1]
(b) Q is the point (10, 10). a and b are natural numbers.
(i) Use your answers to part (a) to find all the possible areas of the rhombus.
..................................................................................................................................................... [4]
(ii) What is the mathematical name of the shape when the rhombus has an area of 100?
......................................................... [2]
518
567
541
