easy geometry

1)If diagonals of a rhombus are 10 cm and 24 cm. find the area and perimeter of the rhombus.
2)A regular hexagon with a perimeter of 24 units is inscribed in a circle. Find the radius of the circle.
3)Find the altitude,perimeter and area of an isosceles trapezoid whose sides have lengths 10 cm, 20 cm, 10 cm and 30 cm.
4) The consecutive sides of a quadrilateral measure (x-17), (24-x), (3x-40) and (x+1). the perimeter is 42 cm. Is the quadrilateral a parallelogram?
5)The perimeter of parallelogram QUAD is 10 more than 5 times AU. If QU = 26 cm. Find AU.

Thanks for those who will share their solutions. God bless!

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  1. 1. In a rhombus, the diagonals bisect each other at right angles. So you have 4 congruent right-angled triangles with sides 5 and 12, and the hypotenuse is the side of the rhombus.
    s^2 = 5^2 + 12^2
    (You should recognize that Pythagorean triple, if not, remember it)
    Find the area of one triangle, then multiply by 4

    2. There are 6 identical isosceles triangle with a base of 24/6 or 4 units, and a central angle of 60°
    The length of those sides is your radius.

    3. I drew a figure ABCD , with AD || BC
    AD = 10
    AB = 20
    BC = 30, and
    CD = 10
    From A draw a line AP, so that AP is || to DC and P is on BC
    So AP=10 and APCD Is a rhombus (all sides = 10)
    Now look at triangle ABP , it is isosceles with sides 20,20, and 10
    We need an angle, I found angle APB
    cos APB =5/20
    angle APB = appr 75.52°
    So angle APC = 104.48°
    Take it from there.

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  2. x-17 + 24-x + 3x-40 + x+1 = 42
    4x = 74
    x = 18.5 ,
    so the sides are 1.5 , 5.5, 15.5 , and 19.5
    how about doubling each side, creating a "similar" quadrilateral, (angles won't change)

    3 , 11 , 31 , and 39 , mmmh, what do you know about sides of a parallelogram ?

    Make your sketch, label QU = 26 , let AU = x
    we know
    perimtetr = x + 26 + x + 26
    = 2x + 52
    but 2x + 52 = 5x + 10
    3x = 42
    x = 14

    So AU = 14 cm

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  3. Thanks Mr.Reiny! But I don't understand your solution in number 3. Can you show another, please.

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