Quadrilateral ABCD≅LMNO≅STUV
. If STUV
is a rectangle and diagonal SU¯¯¯¯¯¯¯=5 cm
, what is the length of diagonals AC¯¯¯¯¯¯¯¯
and BD¯¯¯¯¯¯¯¯
?
Since ABCD is congruent to LMNO and STUV, all corresponding sides and angles are congruent.
Since STUV is a rectangle, opposite sides are parallel and congruent. Since ABCD is congruent to STUV, it is also a rectangle. Therefore, opposite sides of ABCD are parallel and congruent.
We know that AC is a diagonal. In a rectangle, diagonals bisect each other and are equal in length. Therefore, AC is equal to half of the length of BD.
Since ABCD is a rectangle, opposite sides are parallel and congruent. Additionally, the length of AC is equal to the length of BD divided by 2.
Since SU is a diagonal of STUV, it is also equal to the length of AC. Therefore, SU is equal to the length of BD divided by 2.
Given that SU is 5 cm, we can solve for the length of BD:
5 = BD/2
10 = BD
Therefore, the length of BD is 10 cm.
Since AC is equal to half of the length of BD, AC is equal to 10/2 = 5 cm.
Therefore, the length of diagonals AC and BD is 5 cm.
To find the lengths of diagonals AC and BD, we first need to understand the properties of congruent quadrilaterals and the rectangle STUV.
Since quadrilateral ABCD is congruent to quadrilateral LMNO and STUV, we can deduce that all corresponding sides and angles of these quadrilaterals are congruent.
Since STUV is a rectangle, it has the following properties:
1. Opposite sides are parallel and congruent.
2. All angles are right angles (90 degrees).
3. Diagonals are congruent and bisect each other.
Given that diagonal SU is 5 cm, we can deduce that diagonal TV is also 5 cm, as the diagonals of a rectangle are congruent.
Since TUVS is a rectangle, diagonal TV bisects diagonal AC at point X. Thus, AX = XC.
To find the length of diagonal AC, we can use the Pythagorean theorem.
Let's denote the length of AX (or XC) as x. Since SV and UT are parallel sides of the rectangle, AX will be the height of the rectangle.
Using the Pythagorean theorem in right triangle TSV, we have:
TS² + SV² = TV²
TS² + 5² = 5²
TS² + 25 = 25
TS² = 0
Since the sum of the squares of TS and SV is equal to 25, we can conclude that TS has a length of 0. This means that the rectangle STUV degenerates into a line segment.
Therefore, the length of diagonal AC is also 0. Since the diagonals of a rectangle are congruent, the length of diagonal BD is also 0.
So, the length of diagonals AC and BD is 0 cm.
Since quadrilateral ABCD is congruent to quadrilateral LMNO and STUV, it means that all corresponding sides and angles are congruent.
Given that STUV is a rectangle and SU = 5 cm, we can conclude that ST = UV = 5 cm. Also, since STUV is a rectangle, opposite sides are parallel, and the diagonals are congruent. Therefore, STUV is a square.
Now, let's consider the diagonals of the square STUV. Since it is a square, its diagonals are perpendicular bisectors of each other, and they intersect at right angles.
Let's denote the intersection point of diagonals AC and BD as point E.
By the properties of a square, we know that diagonals divide a square into four congruent right triangles.
So, in triangle ASU, we can use the Pythagorean theorem to find the length of the diagonal AC:
AC^2 = AS^2 + SU^2
AC^2 = 5^2 + 5^2
AC^2 = 50
AC = sqrt(50)
AC ≈ 7.07 cm
Similarly, in triangle BST, we can use the Pythagorean theorem to find the length of the diagonal BD:
BD^2 = BS^2 + ST^2
BD^2 = 5^2 + 5^2
BD^2 = 50
BD = sqrt(50)
BD ≈ 7.07 cm
Therefore, the length of diagonals AC and BD is approximately 7.07 cm.